2818:
3277:
2408:
12199:
1197:. Mathematical descriptions of change use differentials and derivatives. Various differentials, derivatives, and functions become related via equations, such that a differential equation is a result that describes dynamically changing phenomena, evolution, and variation. Often, quantities are defined as the rate of change of other quantities (for example, derivatives of displacement with respect to time), or
2951:
2813:{\displaystyle {\begin{pmatrix}y_{1}^{(n)}\\y_{2}^{(n)}\\\vdots \\y_{m}^{(n)}\end{pmatrix}}={\begin{pmatrix}f_{1}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\f_{2}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\\\vdots \\f_{m}\left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)\end{pmatrix}}}
9402:
6186:
4277:, who studied them in the mid-1800s. SLPs have an infinite number of eigenvalues, and the corresponding eigenfunctions form a complete, orthogonal set, which makes orthogonal expansions possible. This is a key idea in applied mathematics, physics, and engineering. SLPs are also useful in the analysis of certain partial differential equations.
8608:
3272:{\displaystyle {\begin{pmatrix}f_{1}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\f_{2}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\\\vdots \\f_{m}(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)})\end{pmatrix}}={\begin{pmatrix}0\\0\\\vdots \\0\end{pmatrix}}}
5578:
1158:). When physical phenomena are modeled with non-linear equations, they are generally approximated by linear differential equations for an easier solution. The few non-linear ODEs that can be solved explicitly are generally solved by transforming the equation into an equivalent linear ODE (see, for example
12260:
differential equation, and then plug the solution into the differential equation to validate if it satisfies the equation. If it does then we have a particular solution to the DE, otherwise we start over again and try another guess. For instance we could guess that the solution to a DE has the form:
4208:
Lie's group theory of differential equations has been certified, namely: (1) that it unifies the many ad hoc methods known for solving differential equations, and (2) that it provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations.
12302:
In the case of a first order ODE that is non-homogeneous we need to first find a solution to the homogeneous portion of the DE, otherwise known as the associated homogeneous equation, and then find a solution to the entire non-homogeneous equation by guessing. Finally, we add both of these solutions
4177:. As the latter can be classified according to the properties of the fundamental curve that remains unchanged under a rational transformation, Clebsch proposed to classify the transcendent functions defined by differential equations according to the invariant properties of the corresponding surfaces
9170:
12259:
When all other methods for solving an ODE fail, or in the cases where we have some intuition about what the solution to a DE might look like, it is sometimes possible to solve a DE simply by guessing the solution and validating it is correct. To use this method, we simply guess a solution to the
6906:
4146:
was the first to appreciate the importance of this view. Thereafter, the real question was no longer whether a solution is possible by means of known functions or their integrals, but whether a given differential equation suffices for the definition of a function of the independent variable or
3882:
For non-linear autonomous ODEs it is possible under some conditions to develop solutions of finite duration, meaning here that from its own dynamics, the system will reach the value zero at an ending time and stays there in zero forever after. These finite-duration solutions can't be analytical
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in order to call this an implicit ODE ; an implicit ODE system satisfying this
Jacobian non-singularity condition can be transformed into an explicit ODE system. In the same sources, implicit ODE systems with a singular Jacobian are termed
951:
9146:
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8712:
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7355:
7184:
12164:
9397:{\displaystyle {\begin{aligned}F(x,y)=&\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda +\int ^{y}Y(\lambda )\,d\lambda \\=&\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda +\int ^{x}X(\lambda )\,d\lambda =C\end{aligned}}}
9047:
5893:
3424:(DAEs). This distinction is not merely one of terminology; DAEs have fundamentally different characteristics and are generally more involved to solve than (nonsingular) ODE systems. Presumably for additional derivatives, the
8403:
7670:
6669:
1905:
8152:
7861:
1387:
5108:
4105:
was a subject of research from the time of
Leibniz, but only since the middle of the nineteenth century has it received special attention. A valuable but little-known work on the subject is that of Houtain (1854).
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12514:
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but this would lead to a domain that is not an interval, so that the side opposite to the initial condition would be disconnected from the initial condition, and therefore not uniquely determined by it.
3883:
functions on the whole real line, and because they will be non-Lipschitz functions at their ending time, they are not included in the uniqueness theorem of solutions of
Lipschitz differential equations.
1650:
10710:
10296:
5033:
3529:
2257:
11854:
5746:
11238:
2833:
8603:{\displaystyle {\begin{aligned}F(x,y)&=\int ^{x}M(\lambda ,y)\,d\lambda +\int ^{y}Y(\lambda )\,d\lambda \\&=\int ^{y}N(x,\lambda )\,d\lambda +\int ^{x}X(\lambda )\,d\lambda =C\end{aligned}}}
11875:
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9409:
9175:
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5573:{\displaystyle {\begin{cases}\mathbb {R} &y_{0}=0\\\left(-\infty ,x_{0}+{\frac {1}{y_{0}}}\right)&y_{0}>0\\\left(x_{0}+{\frac {1}{y_{0}}},+\infty \right)&y_{0}<0\end{cases}}}
11436:
9889:
9738:
10091:
7871:
5147:
4965:
3799:
8829:
8185:
6181:{\displaystyle {\begin{array}{rcl}y_{1}'&=&y_{2}\\y_{2}'&=&y_{3}\\&\vdots &\\y_{n-1}'&=&y_{n}\\y_{n}'&=&F(x,y_{1},\ldots ,y_{n}).\end{array}}}
1123:
8717:
8614:
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3983:
1032:
992:
10307:
4343:
In their basic form both of these theorems only guarantee local results, though the latter can be extended to give a global result, for example, if the conditions of
9073:
1065:
13608:
7260:
7089:
1963:
2180:
3285:
1174:
8992:
6901:{\displaystyle {\begin{aligned}P_{1}(x)Q_{1}(y)+P_{2}(x)Q_{2}(y)\,{\frac {dy}{dx}}&=0\\P_{1}(x)Q_{1}(y)\,dx+P_{2}(x)Q_{2}(y)\,dy&=0\end{aligned}}}
4753:
When the hypotheses of the Picard–Lindelöf theorem are satisfied, then local existence and uniqueness can be extended to a global result. More precisely:
13030:
Elementary
Differential Equations and Boundary Value Problems (4th Edition), W.E. Boyce, R.C. Diprima, Wiley International, John Wiley & Sons, 1986,
12458:
13058:
Mathematical
Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M. R. Spiegel, J. Liu, Schaum's Outline Series, 2009, ISC_2N 978-0-07-154855-7
4193:'s work put the theory of differential equations on a better foundation. He showed that the integration theories of the older mathematicians can, using
8348:
7591:
4729:
solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique. Since there is no restriction on
789:
14003:
4110:(from 1873) was a leader in the theory, and in the geometric interpretation of these solutions he opened a field worked by various writers, notably
14008:
3862:
solution (a general solution of the homogeneous ODE), which then forms a general solution of the original ODE. This is the terminology used in the
708:
8043:
7778:
1300:
13871:
13086:
Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge
University Press, 2010, ISC_2N 978-0-521-86153-3
12220:
449:
17:
11778:
4232:
are used to understand the structure of linear and non-linear (partial) differential equations for generating integrable equations, to find its
13998:
13331:
7071:{\displaystyle \int ^{x}{\frac {P_{1}(\lambda )}{P_{2}(\lambda )}}\,d\lambda +\int ^{y}{\frac {Q_{2}(\lambda )}{Q_{1}(\lambda )}}\,d\lambda =C}
5040:
11132:{\displaystyle y_{c}=C_{1}e^{-{\frac {x}{2}}\,\left(b+{\sqrt {b^{2}-4c}}\right)}+C_{2}e^{-{\frac {x}{2}}\,\left(b-{\sqrt {b^{2}-4c}}\right)}}
4169:. Collet was a prominent contributor beginning in 1869. His method for integrating a non-linear system was communicated to Bertrand in 1868.
6443:{\displaystyle \mathbf {y} =(y_{1},\ldots ,y_{n}),\quad \mathbf {F} (x,y_{1},\ldots ,y_{n})=(y_{2},\ldots ,y_{n},F(x,y_{1},\ldots ,y_{n})).}
5781:
13601:
4257:
Sturm–Liouville theory is a theory of a special type of second-order linear ordinary differential equation. Their solutions are based on
10720:
13658:
12416:, an open-source application that uses a Python-like syntax with a wide range of capabilities spanning several branches of mathematics.
12336:{\displaystyle {\text{general solution}}={\text{general solution of the associated homogeneous equation}}+{\text{particular solution}}}
11450:
192:
6458:
Some differential equations have solutions that can be written in an exact and closed form. Several important classes are given here.
4365:
14127:
12564:
Karras, Tero; Aittala, Miika; Aila, Timo; Laine, Samuli (2022). "Elucidating the Design Space of
Diffusion-Based Generative Models".
6584:
in general). The differential equations are in their equivalent and alternative forms that lead to the solution through integration.
4243:
Symmetry methods have been applied to differential equations that arise in mathematics, physics, engineering, and other disciplines.
4118:. To the latter is due (1872) the theory of singular solutions of differential equations of the first order as accepted circa 1900.
2370:{\displaystyle \mathbf {y} ^{(n)}=\mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n-1)}\right)}
1812:
1688:
14353:
13771:
7700:
7372:
1980:
1556:
323:
3586:
1584:
1479:
10523:
10158:
4974:
3892:
2927:{\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} ',\mathbf {y} '',\ldots ,\mathbf {y} ^{(n)}\right)={\boldsymbol {0}}}
1579:
6197:
13594:
12659:
Denis, Byakatonda (2020-12-10). "An
Overview of Numerical and Analytical Methods for solving Ordinary Differential Equations".
5684:
13815:
11156:
364:
13709:
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13390:
13347:
13284:
12998:
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12880:
12859:
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A tutorial on how to model a physical system described by ODE using Scilab standard programming language by
Openeering team.
5590:
5294:
3453:
13993:
701:
156:
13903:
10099:
3858:
can also refer to any solution of the ODE (not necessarily satisfying the initial conditions), which is then added to the
13918:
13699:
12473:
12188:
11703:
10469:
7194:
3867:
575:
229:
13266:
12453:
9987:{\displaystyle x=\pm \int ^{y}{\frac {d\lambda }{\sqrt {2\int ^{\lambda }F(\varepsilon )\,d\varepsilon +C_{1}}}}+C_{2}}
3851:
is a solution that cannot be obtained by assigning definite values to the arbitrary constants in the general solution.
12438:, an open source computational environment primarily intended for statistics, which includes packages for ODE solving.
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11257:
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4130:. As it had been the hope of eighteenth-century algebraists to find a method for solving the general equation of the
2144:
1138:
Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most
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varies depending upon the author and upon which notation is most useful for the task at hand. In this context, the
318:
237:
212:
8023:{\displaystyle {\begin{aligned}yM(xy)+xN(xy)\,{\frac {dy}{dx}}&=0\\yM(xy)\,dx+xN(xy)\,dy&=0\end{aligned}}}
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13734:
13258:
10026:
5395:
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is derived from the general solution by setting the constants to particular values, often chosen to fulfill set '
3432:, which makes the Jacobian singularity criterion sufficient for this taxonomy to be comprehensive at all orders.
3421:
1915:
1790:
694:
8980:{\displaystyle {\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}}
8336:{\displaystyle {\begin{aligned}M(x,y){\frac {dy}{dx}}+N(x,y)&=0\\M(x,y)\,dy+N(x,y)\,dx&=0\end{aligned}}}
4610:
14348:
14049:
13549:
12224:
12006:{\displaystyle y_{c}=\sum _{j=1}^{n}\left(\sum _{\ell =1}^{k_{j}}C_{j,\ell }x^{\ell -1}\right)e^{\alpha _{j}x}}
5120:
4460:
1661:
629:
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4134:
th degree, so it was the hope of analysts to find a general method for integrating any differential equation.
3746:
354:
14174:
14120:
13820:
13663:
13653:
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13461:
8822:
6587:
In the integral solutions, λ and ε are dummy variables of integration (the continuum analogues of indices in
4102:
746:
369:
197:
187:
9641:{\displaystyle X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}\mu (x,\lambda )N(x,\lambda )\,d\lambda }
9520:{\displaystyle Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}\mu (\lambda ,y)M(\lambda ,y)\,d\lambda }
13673:
13643:
12468:
12407:
4213:
4198:
1935:
1475:
772:
644:
495:
398:
285:
207:
9875:{\displaystyle 2{\frac {dy}{dx}}{\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right)^{2}}
13908:
13778:
13739:
13515:
8178:
330:
245:
7576:{\displaystyle {\begin{aligned}P(y){\frac {dy}{dx}}+Q(x)&=0\\P(y)\,dy+Q(x)\,dx&=0\end{aligned}}}
5908:
4344:
1078:
13099:
4887:{\displaystyle I_{\max }=(x_{-},x_{+}),x_{\pm }\in \mathbb {R} \cup \{\pm \infty \},x_{0}\in I_{\max }}
4266:
4252:
535:
403:
13744:
11575:
10849:
14029:
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12263:
4898:
4166:
1552:
506:
484:
14084:
14054:
4142:. Hence, analysts began to substitute the study of functions, thus opening a new and fertile field.
3430:
any ODE of order greater than one can be (and usually is) rewritten as system of ODEs of first order
649:
14113:
13943:
13457:
12209:
11636:
10910:
5760:
4127:
3408:{\displaystyle {\frac {\partial \mathbf {F} (x,\mathbf {u} ,\mathbf {v} )}{\partial \mathbf {v} }}}
500:
408:
5659:
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5162:
4354:
systems, which may have multiple solutions stemming from their (non-linear) algebraic part alone.
4077:{\displaystyle y(x)={\frac {1}{4}}\left(1-{\frac {x}{2}}+\left|1-{\frac {x}{2}}\right|\right)^{2}}
13953:
13923:
13913:
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13364:
12355:
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This shows clearly that the maximum interval may depend on the initial conditions. The domain of
4308:
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A general solution approach uses the symmetry property of differential equations, the continuous
3871:
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518:
359:
10456:{\displaystyle {\frac {d^{2}y}{dx^{2}}}+2p(x){\frac {dy}{dx}}+\left(p(x)^{2}+p'(x)\right)y=q(x)}
1247:
Many mathematicians have studied differential equations and contributed to the field, including
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4197:, be referred to a common source, and that ordinary differential equations that admit the same
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9141:{\displaystyle {\frac {\partial (\mu M)}{\partial y}}={\frac {\partial (\mu N)}{\partial x}}}
4286:
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734:
639:
624:
513:
456:
438:
277:
33:
12593:. Numerical Analysis 2000. Vol. VI: Ordinary Differential Equations and Integral Equations.
8810:{\displaystyle X(x)=M(x,y)-{\frac {\partial }{\partial x}}\int ^{y}N(x,\lambda )\,d\lambda }
8707:{\displaystyle Y(y)=N(x,y)-{\frac {\partial }{\partial y}}\int ^{x}M(\lambda ,y)\,d\lambda }
4237:
14311:
14216:
13963:
13887:
13856:
13400:
13357:
13162:
13111:
12680:
12598:
11569:
10843:
4330:
4135:
1468:
1209:
525:
461:
434:
13240:
7350:{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=F(y)\\dy&=F(y)\,dx\end{aligned}}}
7179:{\displaystyle {\begin{aligned}{\frac {dy}{dx}}&=F(x)\\dy&=F(x)\,dx\end{aligned}}}
1041:
741:. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the
8:
14306:
14296:
14249:
13973:
13866:
13760:
13376:
13277:
Computer
Methods for Ordinary Differential Equations and Differential-Algebraic Equations
12478:
4362:
The theorem can be stated simply as follows. For the equation and initial value problem:
4314:
4107:
1652:
is often used in physics for representing derivatives of low order with respect to time.
1237:
1151:
1139:
557:
542:
443:
307:
146:
113:
104:
14105:
13380:
12602:
12159:{\displaystyle C_{j}e^{\alpha _{j}x}=C_{j}e^{\chi _{j}x}\cos(\gamma _{j}x+\varphi _{j})}
4126:
The primitive attempt in dealing with differential equations had in view a reduction to
3428:
and so forth are also assumed non-singular according to this scheme, although note that
14328:
14211:
14094:
13933:
13928:
13851:
13683:
12935:
12660:
12565:
12045:
9052:
3339:{\displaystyle \mathbf {F} \left(x,\mathbf {y} ,\mathbf {y} '\right)={\boldsymbol {0}}}
1966:
1948:
1449:
1424:) appears on both sides of the differential equation, and is indicated in the notation
1155:
552:
547:
430:
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13555:
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13465:
13451:
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13414:
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13194:
13184:
13166:
13155:
13137:
13115:
13068:
13031:
12855:
12639:
12616:
12504:
12351:
9042:{\displaystyle {\frac {\partial M}{\partial y}}\neq {\frac {\partial N}{\partial x}}}
4604:
4600:
4098:
3848:
1393:
1252:
1143:
776:
609:
388:
123:
13533:
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5263:
and therefore locally Lipschitz continuous, satisfying the Picard–Lindelöf theorem.
4269:. The problems are identified as Sturm–Liouville problems (SLP) and are named after
664:
14254:
14234:
14079:
14059:
13537:
13335:
13222:
12927:
12606:
12389:
12365:
4285:
There are several theorems that establish existence and uniqueness of solutions to
4274:
4174:
4156:
2166:
1256:
1159:
674:
659:
12299:
since this is a very common solution that physically behaves in a sinusoidal way.
14269:
14206:
14201:
14196:
14039:
13968:
13478:
13406:
13396:
13353:
13325:
13301:
8398:{\displaystyle {\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial x}}}
7665:{\displaystyle \int ^{y}P(\lambda )\,d\lambda +\int ^{x}Q(\lambda )\,d\lambda =C}
5902:
first-order differential equations by defining a new family of unknown functions
3416:
3347:
1194:
614:
530:
57:
14044:
946:{\displaystyle a_{0}(x)y+a_{1}(x)y'+a_{2}(x)y''+\cdots +a_{n}(x)y^{(n)}+b(x)=0,}
669:
14034:
13841:
13321:
13150:
13103:
12919:
12544:
6581:
4201:
present comparable integration difficulties. He also emphasized the subject of
4170:
4139:
3561:
3436:
3425:
1166:
634:
619:
425:
413:
132:
27:
Differential equation containing derivatives with respect to only one variable
14342:
14273:
14229:
13559:
13447:
13133:
12620:
4262:
4162:
4115:
2399:
1268:
1170:
13543:
13530:, containing a list of ordinary differential equations with their solutions.
13339:
12931:
8147:{\displaystyle \ln(Cx)=\int ^{xy}{\frac {N(\lambda )\,d\lambda }{\lambda }}}
7856:{\displaystyle \ln(Cx)=\int ^{y/x}{\frac {d\lambda }{F(\lambda )-\lambda }}}
5266:
Even in such a simple setting, the maximum domain of solution cannot be all
1382:{\displaystyle m{\frac {\mathrm {d} ^{2}x(t)}{\mathrm {d} t^{2}}}=F(x(t))\,}
14089:
14024:
13938:
13552:
from the Holistic Numerical Methods Institute, University of South Florida.
13255:
Applications of Lie's Theory of Ordinary and Partial Differential Equations
12587:"Numerical methods for ordinary differential equations in the 20th century"
12536:
4225:
4221:
1248:
654:
604:
490:
118:
13067:
Further Elementary Analysis, R. Porter, G.Bell & Sons (London), 1978,
5976:-dimensional system of first-order coupled differential equations is then
4173:(1873) attacked the theory along lines parallel to those in his theory of
2227:
A number of coupled differential equations form a system of equations. If
13846:
13586:
13313:
13202:
12398:, a proprietary application primarily intended for symbolic calculations.
12395:
4578:
4258:
2942:
1244:(stock trends, interest rates and the market equilibrium price changes).
1221:
1186:
722:
62:
5103:{\displaystyle \lim _{x\to x_{\pm }}y(x)\ \in \partial {\bar {\Omega }}}
3435:
The behavior of a system of ODEs can be visualized through the use of a
14159:
14074:
12380:, a high-level language, primarily intended for numerical computations.
12377:
11690:
6536:
5751:
which is one of the two possible cases according to the above theorem.
4350:
Also, uniqueness theorems like the Lipschitz one above do not apply to
4217:
4190:
742:
679:
12410:, a high-level language primarily intended for numerical computations.
12048:, allows some terms in the previous results to be written in the form
5888:{\displaystyle F\left(x,y,y',y'',\ \ldots ,\ y^{(n-1)}\right)=y^{(n)}}
4733:
to be linear, this applies to non-linear equations that take the form
13836:
13527:
13429:
CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3
13370:
6588:
4194:
1241:
1225:
1217:
420:
141:
84:
74:
12198:
4357:
4289:
involving ODEs both locally and globally. The two main theorems are
4138:(1799) showed, however, that complex differential equations require
3866:
section in this article, and is frequently used when discussing the
14316:
14154:
14149:
13876:
13568:
An introductory textbook on differential equations by Jiri Lebl of
12665:
12570:
12537:"What is the origin of the term "ordinary differential equations"?"
12500:
A First Course in Differential Equations with Modeling Applications
12413:
4748:
4233:
1205:
1198:
1165:
Some ODEs can be solved explicitly in terms of known functions and
780:
10811:{\displaystyle {\frac {d^{2}y}{dx^{2}}}+b{\frac {dy}{dx}}+cy=r(x)}
4901:
of the solution that satisfies this initial condition with domain
4897:
such that any solution that satisfies this initial condition is a
4770:) there exists a unique maximum (possibly infinite) open interval
12425:
11537:{\displaystyle \sum _{j=0}^{n}b_{j}{\frac {d^{j}y}{dx^{j}}}=r(x)}
1260:
1233:
1229:
1213:
1185:
Ordinary differential equations (ODEs) arise in many contexts of
1147:
745:
of those functions. The term "ordinary" is used in contrast with
95:
90:
79:
13565:
13210:
4165:
inspired a novel approach, subsequently elaborated by Thomé and
13330:, Classics in Applied Mathematics, vol. 38, Philadelphia:
13183:(2nd edition), Chapman & Hall/CRC Press, Boca Raton, 2003.
13181:
Handbook of Exact Solutions for Ordinary Differential Equations
12429:
12383:
12371:
12361:
4147:
variables, and, if so, what are the characteristic properties.
4143:
1201:
of quantities, which is how they enter differential equations.
1190:
756:
one independent variable, and, less commonly, in contrast with
1900:{\displaystyle F\left(x,y,y',y'',\ \ldots ,\ y^{(n)}\right)=0}
1778:{\displaystyle F\left(x,y,y',\ldots ,y^{(n-1)}\right)=y^{(n)}}
13199:
Differential Equations with Applications and Historical Notes
12435:
12419:
12401:
12303:
together to obtain the general solution to the ODE, that is:
7751:{\displaystyle {\frac {dy}{dx}}=F\left({\frac {y}{x}}\right)}
7427:{\displaystyle x=\int ^{y}{\frac {d\lambda }{F(\lambda )}}+C}
2083:{\displaystyle y^{(n)}=\sum _{i=0}^{n-1}a_{i}(x)y^{(i)}+r(x)}
1272:
12368:) software package for the integration and analysis of ODEs.
3672:{\displaystyle F(x,u,u',\ \ldots ,\ u^{(n)})=0\quad x\in I.}
1645:{\displaystyle ({\dot {y}},{\ddot {y}},{\overset {...}{y}})}
1169:. When that is not possible, the equation for computing the
13569:
12464:
List of dynamical systems and differential equations topics
12422:, a Python package that includes an ODE integration module.
10716:
Second-order, linear, inhomogeneous, constant coefficients
10705:{\displaystyle y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left}
10303:
Second-order, linear, inhomogeneous, function coefficients
10291:{\displaystyle y=e^{-\int ^{x}P(\lambda )\,d\lambda }\left}
5566:
5028:{\displaystyle \limsup _{x\to x_{\pm }}\|y(x)\|\to \infty }
3967:{\displaystyle y'=-{\text{sgn}}(y){\sqrt {|y|}},\,\,y(0)=1}
13582:
Solving an ordinary differential equation in Wolfram|Alpha
11849:{\displaystyle y_{c}=\sum _{j=1}^{n}C_{j}e^{\alpha _{j}x}}
10022:
First-order, linear, inhomogeneous, function coefficients
6239:{\displaystyle \mathbf {y} '=\mathbf {F} (x,\mathbf {y} )}
5759:
Differential equations are usually easier to solve if the
1282:
of motion—the relationship between the displacement
14135:
13550:
A primer on analytical solution of differential equations
13366:
A Treatise on Ordinary and Partial Differential Equations
5741:{\displaystyle \lim _{x\to x_{\pm }}\|y(x)\|\to \infty ,}
11233:{\displaystyle y_{c}=(C_{1}x+C_{2})e^{-{\frac {bx}{2}}}}
4280:
779:
in the unknown function and its derivatives, that is an
13566:
Notes on Diffy Qs: Differential Equations for Engineers
12849:
12386:, an open source application for numerical computation.
12374:, a technical computing application (MATrix LABoratory)
12322:
general solution of the associated homogeneous equation
11446:
th-order, linear, inhomogeneous, constant coefficients
4745:), and it can also be applied to systems of equations.
1127:
are the successive derivatives of the unknown function
13483:
Handbook of First Order Partial Differential Equations
13219:
Physics for Scientists and Engineers: Extended version
12563:
12392:, a proprietary application for symbolic calculations.
11706:
5645:{\displaystyle \mathbb {R} \setminus (x_{0}+1/y_{0}),}
5370:{\displaystyle y(x)={\frac {y_{0}}{(x_{0}-x)y_{0}+1}}}
4240:, and finally finding exact analytic solutions to DE.
3524:{\displaystyle F\left(x,y,y',\ldots ,y^{(n)}\right)=0}
3429:
3234:
2960:
2514:
2417:
1173:
of the solutions may be useful. For applied problems,
13556:
Ordinary Differential Equations and Dynamical Systems
13453:
Ordinary Differential Equations and Dynamical Systems
12311:
12266:
12054:
11878:
11781:
11639:
11578:
11453:
11260:
11159:
10972:
10913:
10852:
10723:
10526:
10472:
10310:
10161:
10102:
10029:
9892:
9761:
9680:
9533:
9412:
9173:
9076:
8995:
8832:
8720:
8617:
8416:
8351:
8188:
8046:
7874:
7781:
7703:
7594:
7452:
7375:
7263:
7197:
7092:
6932:
6672:
6258:
6200:
5985:
5911:
5784:
5687:
5662:
5593:
5389:
5297:
5272:
5212:
5165:
5123:
5043:
4977:
4937:
4907:
4779:
4613:
4463:
4368:
3986:
3895:
3749:
3589:
3456:
3355:
3288:
2954:
2836:
2411:
2260:
1983:
1951:
1815:
1691:
1587:
1303:
1175:
numerical methods for ordinary differential equations
1081:
1044:
1004:
964:
792:
12432:, for computing with functions to 15-digit accuracy.
11759:{\textstyle \prod _{j=1}^{n}(\alpha -\alpha _{j})=0}
10145:{\displaystyle e^{\int ^{x}P(\lambda )\,d\lambda }.}
9999:
5766:
2178:
1154:
are solutions of linear differential equations (see
12778:
12776:
12459:
Laplace transform applied to differential equations
12404:, a Python package that can solve ODEs symbolically
10512:{\displaystyle e^{\int ^{x}P(\lambda )\,d\lambda }}
7242:{\displaystyle y=\int ^{x}F(\lambda )\,d\lambda +C}
13305:
13299:
13154:
12739:
12737:
12335:
12291:
12158:
12005:
11848:
11758:
11671:
11601:
11568:substitute and solve polynomial in α, to find the
11536:
11430:
11232:
11131:
10945:
10875:
10842:substitute and solve polynomial in α, to find the
10810:
10704:
10511:
10455:
10290:
10144:
10085:
9986:
9874:
9732:
9640:
9519:
9396:
9140:
9041:
8979:
8809:
8706:
8602:
8397:
8335:
8146:
8022:
7855:
7750:
7664:
7575:
7426:
7349:
7241:
7178:
7070:
6900:
6442:
6238:
6180:
5953:
5887:
5740:
5670:
5644:
5572:
5369:
5280:
5236:
5173:
5141:
5102:
5027:
4959:
4920:
4886:
4707:
4557:
4437:
4076:
3966:
3793:
3671:
3523:
3407:
3338:
3271:
2926:
2812:
2382:explicit system of ordinary differential equations
2369:
2082:
1957:
1899:
1777:
1644:
1381:
1117:
1059:
1026:
986:
945:
14004:List of nonlinear ordinary differential equations
13371:University of Michigan Historical Math Collection
12924:1985 24th IEEE Conference on Decision and Control
9654:
5950:
4358:Local existence and uniqueness theorem simplified
14340:
14009:List of nonlinear partial differential equations
13497:Handbook of Differential Equations (3rd edition)
12983:Logan, J. (2013). Applied mathematics (4th ed.).
12843:
12773:
12591:Journal of Computational and Applied Mathematics
7677:
6635:, without adding constants (explicitly stated).
5689:
5045:
4979:
4913:
4879:
4785:
4749:Global uniqueness and maximum domain of solution
4438:{\displaystyle y'=F(x,y)\,,\quad y_{0}=y(x_{0})}
775:is a differential equation that is defined by a
13098:
12969:harvtxt error: no target: CITEREFLawrence1999 (
12917:
12734:
12716:
11431:{\displaystyle y_{c}=e^{-{\frac {bx}{2}}}\left}
4181:= 0 under rational one-to-one transformations.
3877:
13999:List of linear ordinary differential equations
13332:Society for Industrial and Applied Mathematics
13238:
12852:Surveys in Differential-Algebraic Equations II
12496:
3854:In the context of linear ODE, the terminology
14121:
13602:
13274:
12997:harvtxt error: no target: CITEREFAscher1998 (
12879:harvtxt error: no target: CITEREFAscher1998 (
12835:harvtxt error: no target: CITEREFAscher1998 (
9733:{\displaystyle {\frac {d^{2}y}{dx^{2}}}=F(y)}
6453:
5152:Note that the maximum domain of the solution
4265:of linear operators defined via second-order
3804:A solution that has no extension is called a
1177:can supply an approximation of the solution.
702:
13528:EqWorld: The World of Mathematical Equations
6665:(general case, see below for special cases)
5771:Any explicit differential equation of order
5726:
5711:
5016:
5001:
4855:
4846:
737:(DE) dependent on only a single independent
13411:Lectures on Ordinary Differential Equations
12345:
12227:. Unsourced material may be challenged and
10086:{\displaystyle {\frac {dy}{dx}}+P(x)y=Q(x)}
7765:, then solve by separation of variables in
4121:
3827:th-order equation is a solution containing
2212:A differential equation that is not linear.
14128:
14114:
13616:
13609:
13595:
5156:is always an interval (to have uniqueness)
2231:is a vector whose elements are functions;
1232:(infectious diseases, genetic variation),
709:
695:
13423:
13308:Theory of Ordinary Differential Equations
13026:
13024:
13022:
13020:
13018:
13016:
13014:
13012:
13010:
13008:
12664:
12633:
12610:
12569:
12247:Learn how and when to remove this message
11389:
11328:
11087:
11014:
10650:
10629:
10563:
10500:
10270:
10249:
10198:
10130:
9948:
9631:
9510:
9377:
9345:
9281:
9249:
8956:
8928:
8800:
8697:
8583:
8551:
8506:
8474:
8312:
8284:
8096:
7999:
7971:
7918:
7649:
7617:
7552:
7530:
7336:
7226:
7165:
7055:
6989:
6877:
6829:
6756:
5664:
5595:
5399:
5274:
5167:
5142:{\displaystyle \partial {\bar {\Omega }}}
4839:
4398:
4246:
3945:
3944:
3790:
1793:ordinary differential equation of order n
1378:
766:
13149:
12964:
12905:
12893:
12818:
12782:
12755:
12692:
12175:is an arbitrary constant (phase shift).
4960:{\displaystyle x_{\pm }\neq \pm \infty }
3863:
3794:{\displaystyle u(x)=v(x)\quad x\in I.\,}
1802:ordinary differential equation of order
1294:, is given by the differential equation
763:(SDEs) where the progression is random.
13320:
13252:
13193:
13082:
13080:
13054:
13052:
12794:
12767:
12704:
12584:
12490:
1551:is more useful for differentiation and
14:
14341:
13485:, Taylor & Francis, London, 2002.
13446:
13239:Boscain, Ugo; Chitour, Yacine (2011),
13216:
13127:
13005:
12992:
12874:
12830:
12806:
12743:
12728:
12679:Mathematics for Chemists, D.M. Hirst,
12182:
7253:First-order, autonomous, separable in
6638:
5183:may depend on the specific choice of (
4967:, there are exactly two possibilities
2823:These are not necessarily linear. The
1925:if it does not depend on the variable
14109:
13590:
13534:Online Notes / Differential Equations
12658:
12428:, an open-source package, written in
9166:can be found in a suitable way, then
5754:
4457:are continuous in a closed rectangle
4281:Existence and uniqueness of solutions
4092:
3977:Admits the finite duration solution:
3346:, some sources also require that the
2165:. In this case, there is always the "
1655:
1204:Specific mathematical fields include
13994:List of named differential equations
13375:
13275:Ascher, Uri; Petzold, Linda (1998),
13130:Introduction to Mathematical Physics
13077:
13049:
12920:"Finite Time Differential Equations"
12517:from the original on 17 January 2020
12225:adding citations to reliable sources
12192:
2139:, leading to further classification.
1975:; that is, it can be rewritten as
1212:. Scientific fields include much of
1073:that do not need to be linear, and
752:(PDEs) which may be with respect to
157:List of named differential equations
13919:Method of undetermined coefficients
13700:Dependent and independent variables
13481:, V. F. Zaitsev, and A. Moussiaux,
13179:Polyanin, A. D. and V. F. Zaitsev,
12474:Method of undetermined coefficients
12189:Method of undetermined coefficients
9743:Multiply both sides of equation by
8033:Separation of variables (divide by
6911:Separation of variables (divide by
6191:more compactly in vector notation:
3868:method of undetermined coefficients
2186:A linear differential equation is
1910:There are further classifications:
230:Dependent and independent variables
24:
12850:Achim Ilchmann; Timo Reis (2014).
12683:, 1976, (No ISBN) SBN: 333-18172-7
12454:Examples of differential equations
9576:
9572:
9455:
9451:
9129:
9112:
9097:
9080:
9030:
9022:
9007:
8999:
8763:
8759:
8660:
8656:
8386:
8378:
8363:
8355:
5954:{\displaystyle y_{i}=y^{(i-1)}.\!}
5732:
5537:
5433:
5130:
5124:
5091:
5085:
5022:
4954:
4852:
4150:
3394:
3359:
2150:A linear differential equation is
1337:
1312:
1146:functions that are encountered in
1118:{\displaystyle y',\ldots ,y^{(n)}}
25:
14365:
13511:"Differential equation, ordinary"
13503:
13046:Boscain; Chitour 2011, p. 21
11620:method of variation of parameters
10894:method of variation of parameters
5767:Reduction to a first-order system
5599:
5113:where Ω is the open set in which
2216:
2181:Nonhomogeneous (or inhomogeneous)
1452:representing an unknown function
13816:Carathéodory's existence theorem
13385:, Dover Publications, New York,
13369:, John Wiley and Sons, 1913, in
13157:Advanced Engineering Mathematics
12497:Dennis G. Zill (15 March 2012).
12197:
11602:{\displaystyle e^{\alpha _{j}x}}
10876:{\displaystyle e^{\alpha _{j}x}}
6307:
6260:
6229:
6215:
6203:
5763:of the equation can be reduced.
3422:differential algebraic equations
3398:
3385:
3377:
3363:
3332:
3315:
3306:
3290:
3199:
3180:
3167:
3158:
3108:
3089:
3076:
3067:
3024:
3005:
2992:
2983:
2920:
2895:
2876:
2863:
2854:
2838:
2775:
2756:
2743:
2734:
2674:
2655:
2642:
2633:
2580:
2561:
2548:
2539:
2340:
2321:
2308:
2299:
2283:
2263:
2223:System of differential equations
1574:is more useful for representing
365:Carathéodory's existence theorem
272:
14354:Ordinary differential equations
14260:Least-squares spectral analysis
14187:Fundamental theorem of calculus
13576:Modeling with ODEs using Scilab
13499:, Academic Press, Boston, 1997.
13382:Ordinary Differential Equations
13327:Ordinary differential equations
13293:
13259:Institute of Physics Publishing
13061:
13040:
12986:
12977:
12958:
12946:
12911:
12899:
12887:
12868:
12824:
12812:
12800:
12788:
12761:
12749:
12722:
12710:
12636:Ordinary differential equations
12292:{\displaystyle y=Ae^{\alpha t}}
6305:
4402:
4335:local existence and uniqueness
4184:
3808:. A solution defined on all of
3777:
3656:
1682:. Then an equation of the form
18:Ordinary differential equations
12854:. Springer. pp. 104–105.
12698:
12686:
12673:
12652:
12634:Greenberg, Michael D. (2012).
12627:
12578:
12557:
12529:
12153:
12124:
11747:
11728:
11531:
11525:
11202:
11173:
10805:
10799:
10647:
10641:
10626:
10620:
10560:
10554:
10497:
10491:
10450:
10444:
10427:
10421:
10401:
10394:
10360:
10354:
10267:
10261:
10246:
10240:
10195:
10189:
10127:
10121:
10080:
10074:
10062:
10056:
9945:
9939:
9727:
9721:
9655:General second-order equations
9628:
9616:
9610:
9598:
9564:
9552:
9543:
9537:
9507:
9495:
9489:
9477:
9443:
9431:
9422:
9416:
9374:
9368:
9342:
9330:
9324:
9312:
9278:
9272:
9246:
9234:
9228:
9216:
9193:
9181:
9124:
9115:
9092:
9083:
8953:
8941:
8925:
8913:
8893:
8881:
8852:
8840:
8797:
8785:
8751:
8739:
8730:
8724:
8694:
8682:
8648:
8636:
8627:
8621:
8580:
8574:
8548:
8536:
8503:
8497:
8471:
8459:
8436:
8424:
8309:
8297:
8281:
8269:
8249:
8237:
8208:
8196:
8138:
8135:
8129:
8120:
8114:
8108:
8093:
8087:
8062:
8053:
7996:
7987:
7968:
7959:
7915:
7906:
7894:
7885:
7841:
7835:
7797:
7788:
7646:
7640:
7614:
7608:
7549:
7543:
7527:
7521:
7501:
7495:
7466:
7460:
7412:
7406:
7333:
7327:
7304:
7298:
7223:
7217:
7162:
7156:
7133:
7127:
7049:
7043:
7028:
7022:
6983:
6977:
6962:
6956:
6874:
6868:
6855:
6849:
6826:
6820:
6807:
6801:
6753:
6747:
6734:
6728:
6712:
6706:
6693:
6687:
6434:
6431:
6393:
6355:
6349:
6311:
6299:
6267:
6233:
6219:
6168:
6130:
5942:
5930:
5898:can be written as a system of
5880:
5874:
5856:
5844:
5729:
5723:
5717:
5696:
5636:
5602:
5345:
5326:
5307:
5301:
5133:
5094:
5076:
5070:
5052:
5019:
5013:
5007:
4986:
4819:
4793:
4702:
4664:
4658:
4620:
4552:
4514:
4508:
4470:
4432:
4419:
4395:
4383:
4103:partial differential equations
3996:
3990:
3955:
3949:
3935:
3927:
3921:
3915:
3774:
3768:
3759:
3753:
3647:
3642:
3636:
3593:
3505:
3499:
3447:Given a differential equation
3389:
3367:
3215:
3210:
3204:
3148:
3124:
3119:
3113:
3057:
3040:
3035:
3029:
2973:
2906:
2900:
2792:
2780:
2691:
2679:
2597:
2585:
2493:
2487:
2461:
2455:
2436:
2430:
2357:
2345:
2274:
2268:
2142:
2077:
2071:
2060:
2054:
2046:
2040:
1995:
1989:
1881:
1875:
1770:
1764:
1746:
1734:
1662:Order of differential equation
1639:
1588:
1439:
1404:is a function of the position
1375:
1372:
1366:
1360:
1331:
1325:
1110:
1104:
1054:
1048:
1021:
1015:
981:
975:
931:
925:
914:
908:
900:
894:
864:
858:
834:
828:
809:
803:
727:ordinary differential equation
452: / Integral solutions
13:
1:
13462:American Mathematical Society
13092:
12612:10.1016/S0377-0427(00)00455-6
12585:Butcher, J. C. (2000-12-15).
11672:{\displaystyle y=y_{c}+y_{p}}
10946:{\displaystyle y=y_{c}+y_{p}}
7678:General first-order equations
5037:leaves domain of definition:
4271:J. C. F. Sturm
4214:infinitesimal transformations
4199:infinitesimal transformations
2204:
1913:
1290:of an object under the force
1180:
13644:Notation for differentiation
13257:, Bristol and Philadelphia:
13242:Introduction à l'automatique
12717:Halliday & Resnick (1977
12469:Matrix differential equation
12408:Julia (programming language)
5671:{\displaystyle \mathbb {R} }
5281:{\displaystyle \mathbb {R} }
5174:{\displaystyle \mathbb {R} }
4756:For each initial condition (
4607:, then there is an interval
4267:homogeneous linear equations
3878:Solutions of finite duration
3556:is an interval, is called a
3442:
2123:are continuous functions of
1476:notation for differentiation
773:linear differential equation
496:Exponential response formula
242:Coupled / Decoupled
7:
13740:Exact differential equation
13516:Encyclopedia of Mathematics
12442:
6625:the integration substitute
4216:of solutions to solutions (
4087:
1941:A differential equation is
1921:A differential equation is
10:
14370:
13253:Dresner, Lawrence (1999),
13221:(3rd ed.), New York:
13161:(3rd ed.), New York:
13110:(3rd ed.), New York:
12186:
7438:First-order, separable in
7082:First-order, separable in
6657:First-order, separable in
6563:are real given constants;
6454:Summary of exact solutions
5656:The maximum domain is not
5380:which has maximum domain:
4971:explosion in finite time:
4708:{\displaystyle I=\subset }
4250:
4203:transformations of contact
4154:
3886:As example, the equation:
2251:and its derivatives, then
2220:
1659:
1412:) of the particle at time
1240:(population competition),
14325:
14225:
14144:
14050:Józef Maria Hoene-Wroński
14030:Gottfried Wilhelm Leibniz
14017:
13986:
13896:
13829:
13821:Cauchy–Kowalevski theorem
13798:
13791:
13753:
13692:
13631:
13624:
13431:. Providence: CRC-Press.
11689:are the solutions of the
11622:, though for very simple
10896:, though for very simple
10016:
10013:
10010:
9673:Second-order, autonomous
9667:
9664:
9661:
7696:First-order, homogeneous
7690:
7687:
7684:
6651:
6648:
6645:
6580:are arbitrary constants (
4921:{\displaystyle I_{\max }}
4603:, square brackets denote
4558:{\displaystyle R=\times }
3576:-times differentiable on
3282:For a system of the form
1933:
630:Józef Maria Hoene-Wroński
576:Undetermined coefficients
485:Method of characteristics
370:Cauchy–Kowalevski theorem
13944:Finite difference method
13217:Tipler, Paul A. (1991),
13128:Harper, Charlie (1976),
12918:Vardia T. Haimo (1985).
12484:
12346:Software for ODE solving
6606:just means to integrate
5587:could be taken as being
5237:{\displaystyle y'=y^{2}}
4122:Reduction to quadratures
3833:constants of integration
2941:= (0, 0, ..., 0) is the
1576:higher-order derivatives
1071:differentiable functions
1027:{\displaystyle a_{n}(x)}
987:{\displaystyle a_{0}(x)}
355:Picard–Lindelöf theorem
349:Existence and uniqueness
13924:Variation of parameters
13914:Separation of variables
13811:Peano existence theorem
13806:Picard–Lindelöf theorem
13693:Attributes of variables
13340:10.1137/1.9780898719222
12932:10.1109/CDC.1985.268832
12638:. Hoboken, N.J: Wiley.
12356:computer algebra system
11630:) inspection may work.
11547:Complementary function
10904:) inspection may work.
10821:Complementary function
7867:First-order, separable
7361:Separation of variables
4325:Picard–Lindelöf theorem
4309:Peano existence theorem
4236:, recursion operators,
3872:variation of parameters
1416:. The unknown function
1220:(celestial mechanics),
581:Variation of parameters
571:Separation of variables
360:Peano existence theorem
14192:Calculus of variations
14165:Differential equations
14085:Carl David Tolmé Runge
13659:Differential-algebraic
13618:Differential equations
13544:Differential Equations
13413:, Dover Publications,
12926:. pp. 1729–1733.
12449:Boundary value problem
12337:
12293:
12160:
12022:complex, then setting
12007:
11945:
11912:
11850:
11815:
11760:
11727:
11673:
11603:
11538:
11474:
11432:
11234:
11133:
10947:
10877:
10812:
10706:
10513:
10457:
10292:
10146:
10087:
10011:Differential equation
9988:
9884:then integrate twice.
9876:
9734:
9662:Differential equation
9642:
9521:
9398:
9142:
9043:
8981:
8811:
8708:
8604:
8408:Integrate throughout.
8399:
8337:
8148:
8024:
7857:
7752:
7685:Differential equation
7666:
7586:Integrate throughout.
7577:
7428:
7351:
7243:
7180:
7072:
6902:
6646:Differential equation
6444:
6240:
6182:
5955:
5889:
5742:
5672:
5646:
5574:
5371:
5288:since the solution is
5282:
5238:
5175:
5143:
5104:
5029:
4961:
4922:
4888:
4709:
4559:
4439:
4287:initial value problems
4253:Sturm–Liouville theory
4247:Sturm–Liouville theory
4078:
3968:
3831:arbitrary independent
3795:
3673:
3525:
3409:
3340:
3273:
2928:
2814:
2371:
2245:vector-valued function
2084:
2029:
1969:of the derivatives of
1959:
1901:
1779:
1646:
1383:
1119:
1061:
1028:
988:
947:
767:Differential equations
761:differential equations
750:differential equations
650:Carl David Tolmé Runge
193:Differential-algebraic
34:Differential equations
14349:Differential calculus
14285:Representation theory
14244:quaternionic analysis
14240:Hypercomplex analysis
14138:mathematical analysis
14070:Augustin-Louis Cauchy
14065:Joseph-Louis Lagrange
13959:Finite element method
13949:Crank–Nicolson method
13883:Numerical integration
13862:Exponential stability
13754:Relation to processes
13639:Differential operator
13546:, S.O.S. Mathematics.
13300:Coddington, Earl A.;
12541:hsm.stackexchange.com
12338:
12294:
12161:
12008:
11918:
11892:
11851:
11795:
11761:
11707:
11674:
11604:
11539:
11454:
11433:
11235:
11134:
10948:
10878:
10813:
10707:
10514:
10458:
10293:
10147:
10088:
9989:
9877:
9735:
9643:
9522:
9399:
9143:
9044:
8982:
8812:
8709:
8605:
8400:
8338:
8149:
8025:
7858:
7753:
7667:
7578:
7429:
7352:
7244:
7181:
7073:
6903:
6445:
6241:
6183:
5956:
5890:
5743:
5673:
5647:
5575:
5372:
5283:
5239:
5176:
5144:
5105:
5030:
4962:
4923:
4889:
4710:
4560:
4440:
4345:Grönwall's inequality
4319:local existence only
4230:differential geometry
4079:
3969:
3796:
3674:
3526:
3410:
3341:
3274:
2929:
2815:
2372:
2085:
2003:
1960:
1902:
1780:
1678:, and derivatives of
1647:
1392:which constrains the
1384:
1120:
1062:
1029:
989:
948:
735:differential equation
640:Augustin-Louis Cauchy
625:Joseph-Louis Lagrange
457:Numerical integration
439:Exponential stability
302:Relation to processes
14217:Table of derivatives
13964:Finite volume method
13888:Dirac delta function
13857:Asymptotic stability
13799:Existence/uniqueness
13664:Integro-differential
12503:. Cengage Learning.
12366:Artistic License 2.0
12309:
12264:
12221:improve this section
12052:
11876:
11779:
11704:
11637:
11611:Particular integral
11576:
11570:linearly independent
11451:
11258:
11157:
10970:
10911:
10885:Particular integral
10850:
10844:linearly independent
10721:
10524:
10470:
10466:Integrating factor:
10308:
10159:
10100:
10096:Integrating factor:
10027:
9890:
9759:
9678:
9531:
9410:
9171:
9074:
8993:
8830:
8823:Inexact differential
8718:
8615:
8414:
8349:
8186:
8044:
7872:
7779:
7701:
7592:
7450:
7373:
7261:
7195:
7189:Direct integration.
7090:
6930:
6670:
6591:), and the notation
6461:In the table below,
6256:
6198:
5983:
5909:
5782:
5685:
5660:
5591:
5387:
5295:
5270:
5210:
5163:
5159:may be smaller than
5121:
5041:
4975:
4935:
4905:
4777:
4611:
4461:
4366:
4331:Lipschitz continuous
3984:
3893:
3747:
3682:Given two solutions
3587:
3454:
3353:
3286:
2952:
2834:
2409:
2258:
1981:
1965:can be written as a
1949:
1813:
1689:
1585:
1469:independent variable
1394:motion of a particle
1301:
1278:A simple example is
1224:(weather modeling),
1210:analytical mechanics
1079:
1060:{\displaystyle b(x)}
1042:
1002:
962:
790:
462:Dirac delta function
198:Integro-differential
14297:Continuous function
14250:Functional analysis
13974:Perturbation theory
13954:Runge–Kutta methods
13934:Integral transforms
13867:Rate of convergence
13763:(discrete analogue)
12603:2000JCoAM.125....1B
12479:Recurrence relation
12330:particular solution
12183:The guessing method
6639:Separable equations
6119:
6085:
6036:
6002:
3856:particular solution
3845:boundary conditions
3837:particular solution
2497:
2465:
2440:
1798:More generally, an
1557:Lagrange's notation
1280:Newton's second law
1238:population modeling
1152:applied mathematics
558:Perturbation theory
553:Integral transforms
444:Rate of convergence
310:(discrete analogue)
147:Population dynamics
114:Continuum mechanics
105:Applied mathematics
14329:Mathematics portal
14212:Lists of integrals
14095:Sofya Kovalevskaya
13929:Integrating factor
13852:Lyapunov stability
13772:Stochastic partial
13425:Ibragimov, Nail H.
13195:Simmons, George F.
12333:
12289:
12156:
12003:
11846:
11756:
11669:
11599:
11534:
11428:
11230:
11129:
10943:
10873:
10808:
10702:
10509:
10453:
10288:
10142:
10083:
10004:th order equations
9984:
9872:
9730:
9638:
9517:
9394:
9392:
9138:
9053:Integration factor
9039:
8977:
8975:
8807:
8704:
8600:
8598:
8395:
8333:
8331:
8179:Exact differential
8164:, the solution is
8144:
8020:
8018:
7853:
7748:
7662:
7573:
7571:
7424:
7347:
7345:
7239:
7176:
7174:
7068:
6898:
6896:
6440:
6236:
6178:
6176:
6107:
6067:
6024:
5990:
5951:
5885:
5755:Reduction of order
5738:
5710:
5668:
5642:
5570:
5565:
5367:
5278:
5234:
5171:
5139:
5100:
5066:
5025:
5000:
4957:
4918:
4884:
4705:
4555:
4435:
4261:and corresponding
4238:Bäcklund transform
4099:singular solutions
4093:Singular solutions
4074:
3964:
3841:initial conditions
3791:
3669:
3521:
3405:
3336:
3269:
3263:
3220:
2924:
2810:
2804:
2500:
2477:
2445:
2420:
2367:
2080:
1967:linear combination
1955:
1897:
1775:
1656:General definition
1642:
1480:Leibniz's notation
1450:dependent variable
1444:In what follows,
1379:
1228:(reaction rates),
1156:Holonomic function
1115:
1057:
1024:
984:
943:
548:Integrating factor
389:Initial conditions
324:Stochastic partial
14336:
14335:
14302:Special functions
14265:Harmonic analysis
14103:
14102:
13982:
13981:
13787:
13786:
13558:lecture notes by
13536:by Paul Dawkins,
13471:978-0-8218-8328-0
13392:978-0-486-60349-0
13349:978-0-89871-510-1
13286:978-1-61197-139-2
12861:978-3-319-11050-9
12731:, pp. 78–83)
12645:978-1-118-23002-2
12510:978-1-285-40110-2
12354:, an open-source
12331:
12323:
12315:
12257:
12256:
12249:
12180:
12179:
11618:: in general the
11517:
11416:
11412:
11355:
11351:
11295:
11226:
11120:
11085:
11047:
11012:
10892:: in general the
10782:
10756:
10381:
10343:
10048:
10017:General solution
9997:
9996:
9969:
9968:
9860:
9835:
9817:
9783:
9713:
9668:General solution
9652:
9651:
9583:
9462:
9136:
9104:
9037:
9014:
8873:
8770:
8667:
8393:
8370:
8229:
8142:
7937:
7851:
7742:
7722:
7691:General solution
7675:
7674:
7487:
7416:
7286:
7115:
7053:
6987:
6775:
6652:General solution
5838:
5829:
5688:
5529:
5467:
5365:
5149:is its boundary.
5136:
5097:
5081:
5044:
4978:
4931:In the case that
4601:Cartesian product
4339:
4338:
4275:J. Liouville
4175:Abelian integrals
4056:
4032:
4010:
3939:
3913:
3849:singular solution
3630:
3621:
3403:
2945:. In matrix form
1958:{\displaystyle F}
1869:
1860:
1637:
1615:
1600:
1580:Newton's notation
1396:of constant mass
1352:
777:linear polynomial
719:
718:
610:Gottfried Leibniz
501:Finite difference
293:
292:
154:
153:
124:Dynamical systems
16:(Redirected from
14361:
14255:Fourier analysis
14235:Complex analysis
14136:Major topics in
14130:
14123:
14116:
14107:
14106:
14080:Phyllis Nicolson
14060:Rudolf Lipschitz
13897:Solution methods
13872:Series solutions
13796:
13795:
13629:
13628:
13611:
13604:
13597:
13588:
13587:
13538:Lamar University
13524:
13475:
13442:
13403:
13360:
13317:
13311:
13302:Levinson, Norman
13289:
13271:
13249:
13247:
13235:
13223:Worth Publishers
13213:
13175:
13160:
13146:
13124:
13087:
13084:
13075:
13065:
13059:
13056:
13047:
13044:
13038:
13028:
13003:
13002:
12990:
12984:
12981:
12975:
12974:
12962:
12956:
12950:
12944:
12943:
12915:
12909:
12903:
12897:
12891:
12885:
12884:
12872:
12866:
12865:
12847:
12841:
12840:
12828:
12822:
12816:
12810:
12804:
12798:
12792:
12786:
12780:
12771:
12765:
12759:
12753:
12747:
12741:
12732:
12726:
12720:
12714:
12708:
12707:, pp. 1, 2)
12702:
12696:
12690:
12684:
12677:
12671:
12670:
12668:
12656:
12650:
12649:
12631:
12625:
12624:
12614:
12582:
12576:
12575:
12573:
12561:
12555:
12554:
12552:
12551:
12533:
12527:
12526:
12524:
12522:
12494:
12342:
12340:
12339:
12334:
12332:
12329:
12324:
12321:
12316:
12314:general solution
12313:
12298:
12296:
12295:
12290:
12288:
12287:
12252:
12245:
12241:
12238:
12232:
12201:
12193:
12165:
12163:
12162:
12157:
12152:
12151:
12136:
12135:
12117:
12116:
12112:
12111:
12097:
12096:
12084:
12083:
12079:
12078:
12064:
12063:
12012:
12010:
12009:
12004:
12002:
12001:
11997:
11996:
11982:
11978:
11977:
11976:
11961:
11960:
11944:
11943:
11942:
11932:
11911:
11906:
11888:
11887:
11855:
11853:
11852:
11847:
11845:
11844:
11840:
11839:
11825:
11824:
11814:
11809:
11791:
11790:
11765:
11763:
11762:
11757:
11746:
11745:
11726:
11721:
11678:
11676:
11675:
11670:
11668:
11667:
11655:
11654:
11608:
11606:
11605:
11600:
11598:
11597:
11593:
11592:
11567:
11543:
11541:
11540:
11535:
11518:
11516:
11515:
11514:
11501:
11497:
11496:
11486:
11484:
11483:
11473:
11468:
11437:
11435:
11434:
11429:
11427:
11423:
11422:
11418:
11417:
11411:
11410:
11392:
11391:
11374:
11373:
11361:
11357:
11356:
11350:
11349:
11331:
11330:
11313:
11312:
11298:
11297:
11296:
11291:
11283:
11270:
11269:
11251:
11239:
11237:
11236:
11231:
11229:
11228:
11227:
11222:
11214:
11201:
11200:
11185:
11184:
11169:
11168:
11150:
11138:
11136:
11135:
11130:
11128:
11127:
11126:
11122:
11121:
11110:
11109:
11100:
11086:
11078:
11068:
11067:
11055:
11054:
11053:
11049:
11048:
11037:
11036:
11027:
11013:
11005:
10995:
10994:
10982:
10981:
10963:
10952:
10950:
10949:
10944:
10942:
10941:
10929:
10928:
10882:
10880:
10879:
10874:
10872:
10871:
10867:
10866:
10841:
10817:
10815:
10814:
10809:
10783:
10781:
10773:
10765:
10757:
10755:
10754:
10753:
10740:
10736:
10735:
10725:
10711:
10709:
10708:
10703:
10701:
10697:
10696:
10695:
10680:
10679:
10661:
10657:
10637:
10636:
10616:
10615:
10601:
10600:
10586:
10585:
10571:
10570:
10550:
10549:
10518:
10516:
10515:
10510:
10508:
10507:
10487:
10486:
10462:
10460:
10459:
10454:
10434:
10430:
10420:
10409:
10408:
10382:
10380:
10372:
10364:
10344:
10342:
10341:
10340:
10327:
10323:
10322:
10312:
10297:
10295:
10294:
10289:
10287:
10283:
10257:
10256:
10236:
10235:
10221:
10220:
10206:
10205:
10185:
10184:
10154:Armour formula:
10151:
10149:
10148:
10143:
10138:
10137:
10117:
10116:
10092:
10090:
10089:
10084:
10049:
10047:
10039:
10031:
10014:Solution method
10008:
10007:
9993:
9991:
9990:
9985:
9983:
9982:
9970:
9967:
9966:
9935:
9934:
9922:
9921:
9913:
9911:
9910:
9883:
9881:
9879:
9878:
9873:
9871:
9870:
9865:
9861:
9859:
9851:
9843:
9836:
9834:
9823:
9818:
9816:
9815:
9814:
9801:
9797:
9796:
9786:
9784:
9782:
9774:
9766:
9753:
9739:
9737:
9736:
9731:
9714:
9712:
9711:
9710:
9697:
9693:
9692:
9682:
9665:Solution method
9659:
9658:
9647:
9645:
9644:
9639:
9594:
9593:
9584:
9582:
9571:
9526:
9524:
9523:
9518:
9473:
9472:
9463:
9461:
9450:
9403:
9401:
9400:
9395:
9393:
9364:
9363:
9308:
9307:
9268:
9267:
9212:
9211:
9165:
9147:
9145:
9144:
9139:
9137:
9135:
9127:
9110:
9105:
9103:
9095:
9078:
9068:
9048:
9046:
9045:
9040:
9038:
9036:
9028:
9020:
9015:
9013:
9005:
8997:
8986:
8984:
8983:
8978:
8976:
8874:
8872:
8864:
8856:
8816:
8814:
8813:
8808:
8781:
8780:
8771:
8769:
8758:
8713:
8711:
8710:
8705:
8678:
8677:
8668:
8666:
8655:
8609:
8607:
8606:
8601:
8599:
8570:
8569:
8532:
8531:
8516:
8493:
8492:
8455:
8454:
8404:
8402:
8401:
8396:
8394:
8392:
8384:
8376:
8371:
8369:
8361:
8353:
8342:
8340:
8339:
8334:
8332:
8230:
8228:
8220:
8212:
8153:
8151:
8150:
8145:
8143:
8141:
8103:
8082:
8080:
8079:
8029:
8027:
8026:
8021:
8019:
7938:
7936:
7928:
7920:
7862:
7860:
7859:
7854:
7852:
7850:
7830:
7822:
7820:
7819:
7815:
7757:
7755:
7754:
7749:
7747:
7743:
7735:
7723:
7721:
7713:
7705:
7688:Solution method
7682:
7681:
7671:
7669:
7668:
7663:
7636:
7635:
7604:
7603:
7582:
7580:
7579:
7574:
7572:
7488:
7486:
7478:
7470:
7433:
7431:
7430:
7425:
7417:
7415:
7401:
7393:
7391:
7390:
7356:
7354:
7353:
7348:
7346:
7287:
7285:
7277:
7269:
7248:
7246:
7245:
7240:
7213:
7212:
7185:
7183:
7182:
7177:
7175:
7116:
7114:
7106:
7098:
7077:
7075:
7074:
7069:
7054:
7052:
7042:
7041:
7031:
7021:
7020:
7010:
7008:
7007:
6988:
6986:
6976:
6975:
6965:
6955:
6954:
6944:
6942:
6941:
6907:
6905:
6904:
6899:
6897:
6867:
6866:
6848:
6847:
6819:
6818:
6800:
6799:
6776:
6774:
6766:
6758:
6746:
6745:
6727:
6726:
6705:
6704:
6686:
6685:
6649:Solution method
6643:
6642:
6634:
6620:
6617:with respect to
6616:
6605:
6579:
6562:
6556:
6550:
6544:
6534:
6519:
6504:
6493:
6482:
6471:
6449:
6447:
6446:
6441:
6430:
6429:
6411:
6410:
6386:
6385:
6367:
6366:
6348:
6347:
6329:
6328:
6310:
6298:
6297:
6279:
6278:
6263:
6245:
6243:
6242:
6237:
6232:
6218:
6210:
6206:
6187:
6185:
6184:
6179:
6177:
6167:
6166:
6148:
6147:
6115:
6102:
6101:
6081:
6063:
6057:
6053:
6052:
6032:
6019:
6018:
5998:
5960:
5958:
5957:
5952:
5946:
5945:
5921:
5920:
5894:
5892:
5891:
5886:
5884:
5883:
5865:
5861:
5860:
5859:
5836:
5827:
5823:
5812:
5747:
5745:
5744:
5739:
5709:
5708:
5707:
5677:
5675:
5674:
5669:
5667:
5651:
5649:
5648:
5643:
5635:
5634:
5625:
5614:
5613:
5598:
5579:
5577:
5576:
5571:
5569:
5568:
5556:
5555:
5544:
5540:
5530:
5528:
5527:
5515:
5510:
5509:
5485:
5484:
5473:
5469:
5468:
5466:
5465:
5453:
5448:
5447:
5414:
5413:
5402:
5376:
5374:
5373:
5368:
5366:
5364:
5357:
5356:
5338:
5337:
5324:
5323:
5314:
5287:
5285:
5284:
5279:
5277:
5247:This means that
5243:
5241:
5240:
5235:
5233:
5232:
5220:
5180:
5178:
5177:
5172:
5170:
5148:
5146:
5145:
5140:
5138:
5137:
5129:
5117:is defined, and
5109:
5107:
5106:
5101:
5099:
5098:
5090:
5079:
5065:
5064:
5063:
5034:
5032:
5031:
5026:
4999:
4998:
4997:
4966:
4964:
4963:
4958:
4947:
4946:
4927:
4925:
4924:
4919:
4917:
4916:
4893:
4891:
4890:
4885:
4883:
4882:
4870:
4869:
4842:
4834:
4833:
4818:
4817:
4805:
4804:
4789:
4788:
4724:
4714:
4712:
4711:
4706:
4695:
4694:
4676:
4675:
4651:
4650:
4632:
4631:
4605:closed intervals
4598:
4594:
4564:
4562:
4561:
4556:
4545:
4544:
4526:
4525:
4501:
4500:
4482:
4481:
4444:
4442:
4441:
4436:
4431:
4430:
4412:
4411:
4376:
4294:
4293:
4157:Frobenius method
4101:of ordinary and
4083:
4081:
4080:
4075:
4073:
4072:
4067:
4063:
4062:
4058:
4057:
4049:
4033:
4025:
4011:
4003:
3973:
3971:
3970:
3965:
3940:
3938:
3930:
3925:
3914:
3911:
3903:
3821:general solution
3806:maximal solution
3800:
3798:
3797:
3792:
3739:
3717:
3699:
3678:
3676:
3675:
3670:
3646:
3645:
3628:
3619:
3615:
3551:
3530:
3528:
3527:
3522:
3514:
3510:
3509:
3508:
3484:
3414:
3412:
3411:
3406:
3404:
3402:
3401:
3392:
3388:
3380:
3366:
3357:
3345:
3343:
3342:
3337:
3335:
3327:
3323:
3322:
3318:
3309:
3293:
3278:
3276:
3275:
3270:
3268:
3267:
3225:
3224:
3214:
3213:
3202:
3187:
3183:
3174:
3170:
3161:
3147:
3146:
3123:
3122:
3111:
3096:
3092:
3083:
3079:
3070:
3056:
3055:
3039:
3038:
3027:
3012:
3008:
2999:
2995:
2986:
2972:
2971:
2933:
2931:
2930:
2925:
2923:
2915:
2911:
2910:
2909:
2898:
2883:
2879:
2870:
2866:
2857:
2841:
2819:
2817:
2816:
2811:
2809:
2808:
2801:
2797:
2796:
2795:
2778:
2763:
2759:
2750:
2746:
2737:
2721:
2720:
2700:
2696:
2695:
2694:
2677:
2662:
2658:
2649:
2645:
2636:
2620:
2619:
2606:
2602:
2601:
2600:
2583:
2568:
2564:
2555:
2551:
2542:
2526:
2525:
2505:
2504:
2496:
2485:
2464:
2453:
2439:
2428:
2376:
2374:
2373:
2368:
2366:
2362:
2361:
2360:
2343:
2328:
2324:
2315:
2311:
2302:
2286:
2278:
2277:
2266:
2200:
2175:
2167:trivial solution
2164:
2135:) is called the
2126:
2122:
2111:
2105:
2089:
2087:
2086:
2081:
2064:
2063:
2039:
2038:
2028:
2017:
1999:
1998:
1974:
1964:
1962:
1961:
1956:
1930:
1906:
1904:
1903:
1898:
1890:
1886:
1885:
1884:
1867:
1858:
1854:
1843:
1806:takes the form:
1784:
1782:
1781:
1776:
1774:
1773:
1755:
1751:
1750:
1749:
1719:
1670:, a function of
1651:
1649:
1648:
1643:
1638:
1636:
1622:
1617:
1616:
1608:
1602:
1601:
1593:
1573:
1550:
1548:
1546:
1545:
1540:
1537:
1525:
1523:
1522:
1517:
1514:
1502:
1500:
1499:
1494:
1491:
1466:
1388:
1386:
1385:
1380:
1353:
1351:
1350:
1349:
1340:
1334:
1321:
1320:
1315:
1308:
1257:Bernoulli family
1195:natural sciences
1160:Riccati equation
1134:
1131:of the variable
1130:
1126:
1124:
1122:
1121:
1116:
1114:
1113:
1089:
1068:
1066:
1064:
1063:
1058:
1035:
1033:
1031:
1030:
1025:
1014:
1013:
995:
993:
991:
990:
985:
974:
973:
952:
950:
949:
944:
918:
917:
893:
892:
874:
857:
856:
844:
827:
826:
802:
801:
711:
704:
697:
675:Phyllis Nicolson
660:Rudolf Lipschitz
543:Green's function
519:Infinite element
510:
475:Solution methods
453:
311:
222:By variable type
176:
175:
58:Natural sciences
51:
50:
30:
29:
21:
14369:
14368:
14364:
14363:
14362:
14360:
14359:
14358:
14339:
14338:
14337:
14332:
14321:
14270:P-adic analysis
14221:
14207:Matrix calculus
14202:Tensor calculus
14197:Vector calculus
14160:Differentiation
14140:
14134:
14104:
14099:
14040:Jacob Bernoulli
14013:
13978:
13969:Galerkin method
13892:
13830:Solution topics
13825:
13783:
13749:
13688:
13620:
13615:
13509:
13506:
13495:D. Zwillinger,
13472:
13439:
13407:Witold Hurewicz
13393:
13377:Ince, Edward L.
13350:
13322:Hartman, Philip
13296:
13287:
13269:
13245:
13233:
13173:
13151:Kreyszig, Erwin
13144:
13122:
13104:Resnick, Robert
13100:Halliday, David
13095:
13090:
13085:
13078:
13066:
13062:
13057:
13050:
13045:
13041:
13029:
13006:
12996:
12991:
12987:
12982:
12978:
12968:
12963:
12959:
12951:
12947:
12916:
12912:
12904:
12900:
12892:
12888:
12878:
12873:
12869:
12862:
12848:
12844:
12834:
12829:
12825:
12817:
12813:
12805:
12801:
12793:
12789:
12781:
12774:
12766:
12762:
12754:
12750:
12742:
12735:
12727:
12723:
12715:
12711:
12703:
12699:
12691:
12687:
12681:Macmillan Press
12678:
12674:
12657:
12653:
12646:
12632:
12628:
12583:
12579:
12562:
12558:
12549:
12547:
12535:
12534:
12530:
12520:
12518:
12511:
12495:
12491:
12487:
12445:
12348:
12328:
12320:
12312:
12310:
12307:
12306:
12280:
12276:
12265:
12262:
12261:
12253:
12242:
12236:
12233:
12218:
12202:
12191:
12185:
12174:
12147:
12143:
12131:
12127:
12107:
12103:
12102:
12098:
12092:
12088:
12074:
12070:
12069:
12065:
12059:
12055:
12053:
12050:
12049:
12046:Euler's formula
12043:
12034:
12021:
11992:
11988:
11987:
11983:
11966:
11962:
11950:
11946:
11938:
11934:
11933:
11922:
11917:
11913:
11907:
11896:
11883:
11879:
11877:
11874:
11873:
11870:
11864:
11835:
11831:
11830:
11826:
11820:
11816:
11810:
11799:
11786:
11782:
11780:
11777:
11776:
11775:all different,
11774:
11741:
11737:
11722:
11711:
11705:
11702:
11701:
11688:
11663:
11659:
11650:
11646:
11638:
11635:
11634:
11616:
11588:
11584:
11583:
11579:
11577:
11574:
11573:
11560:
11555:
11552:
11510:
11506:
11502:
11492:
11488:
11487:
11485:
11479:
11475:
11469:
11458:
11452:
11449:
11448:
11406:
11402:
11390:
11385:
11381:
11369:
11365:
11345:
11341:
11329:
11324:
11320:
11308:
11304:
11303:
11299:
11284:
11282:
11278:
11274:
11265:
11261:
11259:
11256:
11255:
11243:
11215:
11213:
11209:
11205:
11196:
11192:
11180:
11176:
11164:
11160:
11158:
11155:
11154:
11142:
11105:
11101:
11099:
11092:
11088:
11077:
11073:
11069:
11063:
11059:
11032:
11028:
11026:
11019:
11015:
11004:
11000:
10996:
10990:
10986:
10977:
10973:
10971:
10968:
10967:
10955:
10937:
10933:
10924:
10920:
10912:
10909:
10908:
10890:
10862:
10858:
10857:
10853:
10851:
10848:
10847:
10834:
10829:
10826:
10774:
10766:
10764:
10749:
10745:
10741:
10731:
10727:
10726:
10724:
10722:
10719:
10718:
10691:
10687:
10675:
10671:
10611:
10607:
10606:
10602:
10596:
10592:
10591:
10587:
10581:
10577:
10576:
10572:
10545:
10541:
10537:
10533:
10525:
10522:
10521:
10482:
10478:
10477:
10473:
10471:
10468:
10467:
10413:
10404:
10400:
10390:
10386:
10373:
10365:
10363:
10336:
10332:
10328:
10318:
10314:
10313:
10311:
10309:
10306:
10305:
10231:
10227:
10226:
10222:
10216:
10212:
10211:
10207:
10180:
10176:
10172:
10168:
10160:
10157:
10156:
10112:
10108:
10107:
10103:
10101:
10098:
10097:
10040:
10032:
10030:
10028:
10025:
10024:
10006:
9978:
9974:
9962:
9958:
9930:
9926:
9914:
9912:
9906:
9902:
9891:
9888:
9887:
9866:
9852:
9844:
9842:
9838:
9837:
9827:
9822:
9810:
9806:
9802:
9792:
9788:
9787:
9785:
9775:
9767:
9765:
9760:
9757:
9756:
9755:
9744:
9706:
9702:
9698:
9688:
9684:
9683:
9681:
9679:
9676:
9675:
9657:
9589:
9585:
9575:
9570:
9532:
9529:
9528:
9468:
9464:
9454:
9449:
9411:
9408:
9407:
9391:
9390:
9359:
9355:
9303:
9299:
9295:
9289:
9288:
9263:
9259:
9207:
9203:
9199:
9174:
9172:
9169:
9168:
9152:
9128:
9111:
9109:
9096:
9079:
9077:
9075:
9072:
9071:
9055:
9029:
9021:
9019:
9006:
8998:
8996:
8994:
8991:
8990:
8974:
8973:
8963:
8907:
8906:
8896:
8865:
8857:
8855:
8833:
8831:
8828:
8827:
8776:
8772:
8762:
8757:
8719:
8716:
8715:
8673:
8669:
8659:
8654:
8616:
8613:
8612:
8597:
8596:
8565:
8561:
8527:
8523:
8514:
8513:
8488:
8484:
8450:
8446:
8439:
8417:
8415:
8412:
8411:
8385:
8377:
8375:
8362:
8354:
8352:
8350:
8347:
8346:
8330:
8329:
8319:
8263:
8262:
8252:
8221:
8213:
8211:
8189:
8187:
8184:
8183:
8104:
8083:
8081:
8072:
8068:
8045:
8042:
8041:
8017:
8016:
8006:
7950:
7949:
7939:
7929:
7921:
7919:
7875:
7873:
7870:
7869:
7831:
7823:
7821:
7811:
7807:
7803:
7780:
7777:
7776:
7734:
7730:
7714:
7706:
7704:
7702:
7699:
7698:
7680:
7631:
7627:
7599:
7595:
7593:
7590:
7589:
7570:
7569:
7559:
7515:
7514:
7504:
7479:
7471:
7469:
7453:
7451:
7448:
7447:
7402:
7394:
7392:
7386:
7382:
7374:
7371:
7370:
7344:
7343:
7317:
7308:
7307:
7288:
7278:
7270:
7268:
7264:
7262:
7259:
7258:
7208:
7204:
7196:
7193:
7192:
7173:
7172:
7146:
7137:
7136:
7117:
7107:
7099:
7097:
7093:
7091:
7088:
7087:
7037:
7033:
7032:
7016:
7012:
7011:
7009:
7003:
6999:
6971:
6967:
6966:
6950:
6946:
6945:
6943:
6937:
6933:
6931:
6928:
6927:
6923:
6917:
6895:
6894:
6884:
6862:
6858:
6843:
6839:
6814:
6810:
6795:
6791:
6788:
6787:
6777:
6767:
6759:
6757:
6741:
6737:
6722:
6718:
6700:
6696:
6681:
6677:
6673:
6671:
6668:
6667:
6641:
6626:
6618:
6607:
6592:
6577:
6570:
6564:
6558:
6552:
6546:
6540:
6521:
6506:
6495:
6484:
6473:
6462:
6456:
6425:
6421:
6406:
6402:
6381:
6377:
6362:
6358:
6343:
6339:
6324:
6320:
6306:
6293:
6289:
6274:
6270:
6259:
6257:
6254:
6253:
6228:
6214:
6202:
6201:
6199:
6196:
6195:
6175:
6174:
6162:
6158:
6143:
6139:
6125:
6120:
6111:
6104:
6103:
6097:
6093:
6091:
6086:
6071:
6064:
6062:
6055:
6054:
6048:
6044:
6042:
6037:
6028:
6021:
6020:
6014:
6010:
6008:
6003:
5994:
5986:
5984:
5981:
5980:
5929:
5925:
5916:
5912:
5910:
5907:
5906:
5873:
5869:
5843:
5839:
5816:
5805:
5792:
5788:
5783:
5780:
5779:
5769:
5757:
5703:
5699:
5692:
5686:
5683:
5682:
5663:
5661:
5658:
5657:
5630:
5626:
5621:
5609:
5605:
5594:
5592:
5589:
5588:
5564:
5563:
5551:
5547:
5545:
5523:
5519:
5514:
5505:
5501:
5500:
5496:
5493:
5492:
5480:
5476:
5474:
5461:
5457:
5452:
5443:
5439:
5429:
5425:
5422:
5421:
5409:
5405:
5403:
5398:
5391:
5390:
5388:
5385:
5384:
5352:
5348:
5333:
5329:
5325:
5319:
5315:
5313:
5296:
5293:
5292:
5273:
5271:
5268:
5267:
5228:
5224:
5213:
5211:
5208:
5207:
5196:
5189:
5166:
5164:
5161:
5160:
5128:
5127:
5122:
5119:
5118:
5089:
5088:
5059:
5055:
5048:
5042:
5039:
5038:
4993:
4989:
4982:
4976:
4973:
4972:
4942:
4938:
4936:
4933:
4932:
4912:
4908:
4906:
4903:
4902:
4878:
4874:
4865:
4861:
4838:
4829:
4825:
4813:
4809:
4800:
4796:
4784:
4780:
4778:
4775:
4774:
4769:
4762:
4751:
4716:
4690:
4686:
4671:
4667:
4646:
4642:
4627:
4623:
4612:
4609:
4608:
4596:
4582:
4581:(symbolically:
4540:
4536:
4521:
4517:
4496:
4492:
4477:
4473:
4462:
4459:
4458:
4426:
4422:
4407:
4403:
4369:
4367:
4364:
4363:
4360:
4283:
4255:
4249:
4187:
4161:Two memoirs by
4159:
4153:
4151:Fuchsian theory
4140:complex numbers
4124:
4095:
4090:
4068:
4048:
4041:
4037:
4024:
4017:
4013:
4012:
4002:
3985:
3982:
3981:
3934:
3926:
3924:
3910:
3896:
3894:
3891:
3890:
3880:
3864:guessing method
3814:global solution
3748:
3745:
3744:
3731:
3701:
3683:
3635:
3631:
3608:
3588:
3585:
3584:
3535:
3498:
3494:
3477:
3464:
3460:
3455:
3452:
3451:
3445:
3397:
3393:
3384:
3376:
3362:
3358:
3356:
3354:
3351:
3350:
3348:Jacobian matrix
3331:
3314:
3313:
3305:
3298:
3294:
3289:
3287:
3284:
3283:
3262:
3261:
3255:
3254:
3248:
3247:
3241:
3240:
3230:
3229:
3219:
3218:
3203:
3198:
3197:
3179:
3178:
3166:
3165:
3157:
3142:
3138:
3135:
3134:
3128:
3127:
3112:
3107:
3106:
3088:
3087:
3075:
3074:
3066:
3051:
3047:
3044:
3043:
3028:
3023:
3022:
3004:
3003:
2991:
2990:
2982:
2967:
2963:
2956:
2955:
2953:
2950:
2949:
2919:
2899:
2894:
2893:
2875:
2874:
2862:
2861:
2853:
2846:
2842:
2837:
2835:
2832:
2831:
2803:
2802:
2779:
2774:
2773:
2755:
2754:
2742:
2741:
2733:
2726:
2722:
2716:
2712:
2709:
2708:
2702:
2701:
2678:
2673:
2672:
2654:
2653:
2641:
2640:
2632:
2625:
2621:
2615:
2611:
2608:
2607:
2584:
2579:
2578:
2560:
2559:
2547:
2546:
2538:
2531:
2527:
2521:
2517:
2510:
2509:
2499:
2498:
2486:
2481:
2474:
2473:
2467:
2466:
2454:
2449:
2442:
2441:
2429:
2424:
2413:
2412:
2410:
2407:
2406:
2344:
2339:
2338:
2320:
2319:
2307:
2306:
2298:
2291:
2287:
2282:
2267:
2262:
2261:
2259:
2256:
2255:
2225:
2219:
2209:
2191:
2183:
2170:
2155:
2147:
2127:. The function
2124:
2113:
2103:
2102:
2093:
2053:
2049:
2034:
2030:
2018:
2007:
1988:
1984:
1982:
1979:
1978:
1970:
1950:
1947:
1946:
1938:
1926:
1918:
1874:
1870:
1847:
1836:
1823:
1819:
1814:
1811:
1810:
1763:
1759:
1733:
1729:
1712:
1699:
1695:
1690:
1687:
1686:
1664:
1658:
1626:
1621:
1607:
1606:
1592:
1591:
1586:
1583:
1582:
1578:compactly, and
1559:
1541:
1538:
1530:
1529:
1527:
1518:
1515:
1507:
1506:
1504:
1495:
1492:
1487:
1486:
1484:
1482:
1453:
1442:
1345:
1341:
1336:
1335:
1316:
1311:
1310:
1309:
1307:
1302:
1299:
1298:
1183:
1132:
1128:
1103:
1099:
1082:
1080:
1077:
1076:
1074:
1043:
1040:
1039:
1037:
1009:
1005:
1003:
1000:
999:
997:
969:
965:
963:
960:
959:
957:
907:
903:
888:
884:
867:
852:
848:
837:
822:
818:
797:
793:
791:
788:
787:
769:
715:
686:
685:
684:
615:Jacob Bernoulli
599:
586:
585:
567:
536:Petrov–Galerkin
504:
489:
476:
468:
467:
466:
448:
394:Boundary values
383:
375:
374:
350:
337:
336:
335:
309:
303:
295:
294:
282:
259:
217:
173:
160:
159:
155:
133:Social sciences
89:
67:
48:
28:
23:
22:
15:
12:
11:
5:
14367:
14357:
14356:
14351:
14334:
14333:
14326:
14323:
14322:
14320:
14319:
14314:
14309:
14304:
14299:
14294:
14288:
14287:
14282:
14280:Measure theory
14277:
14274:P-adic numbers
14267:
14262:
14257:
14252:
14247:
14237:
14232:
14226:
14223:
14222:
14220:
14219:
14214:
14209:
14204:
14199:
14194:
14189:
14184:
14183:
14182:
14177:
14172:
14162:
14157:
14145:
14142:
14141:
14133:
14132:
14125:
14118:
14110:
14101:
14100:
14098:
14097:
14092:
14087:
14082:
14077:
14072:
14067:
14062:
14057:
14055:Ernst Lindelöf
14052:
14047:
14042:
14037:
14035:Leonhard Euler
14032:
14027:
14021:
14019:
14018:Mathematicians
14015:
14014:
14012:
14011:
14006:
14001:
13996:
13990:
13988:
13984:
13983:
13980:
13979:
13977:
13976:
13971:
13966:
13961:
13956:
13951:
13946:
13941:
13936:
13931:
13926:
13921:
13916:
13911:
13906:
13900:
13898:
13894:
13893:
13891:
13890:
13885:
13880:
13874:
13869:
13864:
13859:
13854:
13849:
13844:
13842:Phase portrait
13839:
13833:
13831:
13827:
13826:
13824:
13823:
13818:
13813:
13808:
13802:
13800:
13793:
13789:
13788:
13785:
13784:
13782:
13781:
13776:
13775:
13774:
13764:
13757:
13755:
13751:
13750:
13748:
13747:
13745:On jet bundles
13742:
13737:
13732:
13727:
13722:
13717:
13712:
13710:Nonhomogeneous
13707:
13702:
13696:
13694:
13690:
13689:
13687:
13686:
13681:
13676:
13671:
13666:
13661:
13656:
13651:
13646:
13641:
13635:
13633:
13626:
13625:Classification
13622:
13621:
13614:
13613:
13606:
13599:
13591:
13585:
13584:
13579:
13573:
13563:
13553:
13547:
13541:
13531:
13525:
13505:
13504:External links
13502:
13501:
13500:
13493:
13479:A. D. Polyanin
13476:
13470:
13448:Teschl, Gerald
13444:
13437:
13421:
13404:
13391:
13373:
13361:
13348:
13318:
13295:
13292:
13291:
13290:
13285:
13272:
13268:978-0750305303
13267:
13250:
13236:
13231:
13214:
13191:
13177:
13171:
13147:
13142:
13132:, New Jersey:
13125:
13120:
13094:
13091:
13089:
13088:
13076:
13060:
13048:
13039:
13004:
12985:
12976:
12965:Lawrence (1999
12957:
12945:
12910:
12906:Kreyszig (1972
12898:
12894:Kreyszig (1972
12886:
12867:
12860:
12842:
12823:
12819:Kreyszig (1972
12811:
12809:, p. 128)
12799:
12787:
12783:Kreyszig (1972
12772:
12760:
12756:Kreyszig (1972
12748:
12746:, p. 127)
12733:
12721:
12709:
12697:
12693:Kreyszig (1972
12685:
12672:
12651:
12644:
12626:
12577:
12556:
12545:Stack Exchange
12528:
12509:
12488:
12486:
12483:
12482:
12481:
12476:
12471:
12466:
12461:
12456:
12451:
12444:
12441:
12440:
12439:
12433:
12423:
12417:
12411:
12405:
12399:
12393:
12387:
12381:
12375:
12369:
12359:
12347:
12344:
12327:
12319:
12286:
12283:
12279:
12275:
12272:
12269:
12255:
12254:
12205:
12203:
12196:
12187:Main article:
12184:
12181:
12178:
12177:
12170:
12155:
12150:
12146:
12142:
12139:
12134:
12130:
12126:
12123:
12120:
12115:
12110:
12106:
12101:
12095:
12091:
12087:
12082:
12077:
12073:
12068:
12062:
12058:
12039:
12030:
12017:
12000:
11995:
11991:
11986:
11981:
11975:
11972:
11969:
11965:
11959:
11956:
11953:
11949:
11941:
11937:
11931:
11928:
11925:
11921:
11916:
11910:
11905:
11902:
11899:
11895:
11891:
11886:
11882:
11868:
11860:
11856:for each root
11843:
11838:
11834:
11829:
11823:
11819:
11813:
11808:
11805:
11802:
11798:
11794:
11789:
11785:
11770:
11755:
11752:
11749:
11744:
11740:
11736:
11733:
11730:
11725:
11720:
11717:
11714:
11710:
11684:
11666:
11662:
11658:
11653:
11649:
11645:
11642:
11632:
11614:
11596:
11591:
11587:
11582:
11558:
11550:
11545:
11533:
11530:
11527:
11524:
11521:
11513:
11509:
11505:
11500:
11495:
11491:
11482:
11478:
11472:
11467:
11464:
11461:
11457:
11440:
11439:
11426:
11421:
11415:
11409:
11405:
11401:
11398:
11395:
11388:
11384:
11380:
11377:
11372:
11368:
11364:
11360:
11354:
11348:
11344:
11340:
11337:
11334:
11327:
11323:
11319:
11316:
11311:
11307:
11302:
11294:
11290:
11287:
11281:
11277:
11273:
11268:
11264:
11225:
11221:
11218:
11212:
11208:
11204:
11199:
11195:
11191:
11188:
11183:
11179:
11175:
11172:
11167:
11163:
11125:
11119:
11116:
11113:
11108:
11104:
11098:
11095:
11091:
11084:
11081:
11076:
11072:
11066:
11062:
11058:
11052:
11046:
11043:
11040:
11035:
11031:
11025:
11022:
11018:
11011:
11008:
11003:
10999:
10993:
10989:
10985:
10980:
10976:
10940:
10936:
10932:
10927:
10923:
10919:
10916:
10906:
10888:
10870:
10865:
10861:
10856:
10832:
10824:
10819:
10807:
10804:
10801:
10798:
10795:
10792:
10789:
10786:
10780:
10777:
10772:
10769:
10763:
10760:
10752:
10748:
10744:
10739:
10734:
10730:
10713:
10712:
10700:
10694:
10690:
10686:
10683:
10678:
10674:
10670:
10667:
10664:
10660:
10656:
10653:
10649:
10646:
10643:
10640:
10635:
10632:
10628:
10625:
10622:
10619:
10614:
10610:
10605:
10599:
10595:
10590:
10584:
10580:
10575:
10569:
10566:
10562:
10559:
10556:
10553:
10548:
10544:
10540:
10536:
10532:
10529:
10519:
10506:
10503:
10499:
10496:
10493:
10490:
10485:
10481:
10476:
10464:
10452:
10449:
10446:
10443:
10440:
10437:
10433:
10429:
10426:
10423:
10419:
10416:
10412:
10407:
10403:
10399:
10396:
10393:
10389:
10385:
10379:
10376:
10371:
10368:
10362:
10359:
10356:
10353:
10350:
10347:
10339:
10335:
10331:
10326:
10321:
10317:
10300:
10299:
10286:
10282:
10279:
10276:
10273:
10269:
10266:
10263:
10260:
10255:
10252:
10248:
10245:
10242:
10239:
10234:
10230:
10225:
10219:
10215:
10210:
10204:
10201:
10197:
10194:
10191:
10188:
10183:
10179:
10175:
10171:
10167:
10164:
10152:
10141:
10136:
10133:
10129:
10126:
10123:
10120:
10115:
10111:
10106:
10094:
10082:
10079:
10076:
10073:
10070:
10067:
10064:
10061:
10058:
10055:
10052:
10046:
10043:
10038:
10035:
10019:
10018:
10015:
10012:
10005:
10000:Linear to the
9998:
9995:
9994:
9981:
9977:
9973:
9965:
9961:
9957:
9954:
9951:
9947:
9944:
9941:
9938:
9933:
9929:
9925:
9920:
9917:
9909:
9905:
9901:
9898:
9895:
9885:
9869:
9864:
9858:
9855:
9850:
9847:
9841:
9833:
9830:
9826:
9821:
9813:
9809:
9805:
9800:
9795:
9791:
9781:
9778:
9773:
9770:
9764:
9741:
9729:
9726:
9723:
9720:
9717:
9709:
9705:
9701:
9696:
9691:
9687:
9670:
9669:
9666:
9663:
9656:
9653:
9650:
9649:
9637:
9634:
9630:
9627:
9624:
9621:
9618:
9615:
9612:
9609:
9606:
9603:
9600:
9597:
9592:
9588:
9581:
9578:
9574:
9569:
9566:
9563:
9560:
9557:
9554:
9551:
9548:
9545:
9542:
9539:
9536:
9516:
9513:
9509:
9506:
9503:
9500:
9497:
9494:
9491:
9488:
9485:
9482:
9479:
9476:
9471:
9467:
9460:
9457:
9453:
9448:
9445:
9442:
9439:
9436:
9433:
9430:
9427:
9424:
9421:
9418:
9415:
9389:
9386:
9383:
9380:
9376:
9373:
9370:
9367:
9362:
9358:
9354:
9351:
9348:
9344:
9341:
9338:
9335:
9332:
9329:
9326:
9323:
9320:
9317:
9314:
9311:
9306:
9302:
9298:
9296:
9294:
9291:
9290:
9287:
9284:
9280:
9277:
9274:
9271:
9266:
9262:
9258:
9255:
9252:
9248:
9245:
9242:
9239:
9236:
9233:
9230:
9227:
9224:
9221:
9218:
9215:
9210:
9206:
9202:
9200:
9198:
9195:
9192:
9189:
9186:
9183:
9180:
9177:
9176:
9149:
9134:
9131:
9126:
9123:
9120:
9117:
9114:
9108:
9102:
9099:
9094:
9091:
9088:
9085:
9082:
9050:
9035:
9032:
9027:
9024:
9018:
9012:
9009:
9004:
9001:
8972:
8969:
8966:
8964:
8962:
8959:
8955:
8952:
8949:
8946:
8943:
8940:
8937:
8934:
8931:
8927:
8924:
8921:
8918:
8915:
8912:
8909:
8908:
8905:
8902:
8899:
8897:
8895:
8892:
8889:
8886:
8883:
8880:
8877:
8871:
8868:
8863:
8860:
8854:
8851:
8848:
8845:
8842:
8839:
8836:
8835:
8825:, first-order
8819:
8818:
8806:
8803:
8799:
8796:
8793:
8790:
8787:
8784:
8779:
8775:
8768:
8765:
8761:
8756:
8753:
8750:
8747:
8744:
8741:
8738:
8735:
8732:
8729:
8726:
8723:
8703:
8700:
8696:
8693:
8690:
8687:
8684:
8681:
8676:
8672:
8665:
8662:
8658:
8653:
8650:
8647:
8644:
8641:
8638:
8635:
8632:
8629:
8626:
8623:
8620:
8595:
8592:
8589:
8586:
8582:
8579:
8576:
8573:
8568:
8564:
8560:
8557:
8554:
8550:
8547:
8544:
8541:
8538:
8535:
8530:
8526:
8522:
8519:
8517:
8515:
8512:
8509:
8505:
8502:
8499:
8496:
8491:
8487:
8483:
8480:
8477:
8473:
8470:
8467:
8464:
8461:
8458:
8453:
8449:
8445:
8442:
8440:
8438:
8435:
8432:
8429:
8426:
8423:
8420:
8419:
8409:
8406:
8391:
8388:
8383:
8380:
8374:
8368:
8365:
8360:
8357:
8328:
8325:
8322:
8320:
8318:
8315:
8311:
8308:
8305:
8302:
8299:
8296:
8293:
8290:
8287:
8283:
8280:
8277:
8274:
8271:
8268:
8265:
8264:
8261:
8258:
8255:
8253:
8251:
8248:
8245:
8242:
8239:
8236:
8233:
8227:
8224:
8219:
8216:
8210:
8207:
8204:
8201:
8198:
8195:
8192:
8191:
8181:, first-order
8175:
8174:
8140:
8137:
8134:
8131:
8128:
8125:
8122:
8119:
8116:
8113:
8110:
8107:
8102:
8099:
8095:
8092:
8089:
8086:
8078:
8075:
8071:
8067:
8064:
8061:
8058:
8055:
8052:
8049:
8038:
8031:
8015:
8012:
8009:
8007:
8005:
8002:
7998:
7995:
7992:
7989:
7986:
7983:
7980:
7977:
7974:
7970:
7967:
7964:
7961:
7958:
7955:
7952:
7951:
7948:
7945:
7942:
7940:
7935:
7932:
7927:
7924:
7917:
7914:
7911:
7908:
7905:
7902:
7899:
7896:
7893:
7890:
7887:
7884:
7881:
7878:
7877:
7864:
7863:
7849:
7846:
7843:
7840:
7837:
7834:
7829:
7826:
7818:
7814:
7810:
7806:
7802:
7799:
7796:
7793:
7790:
7787:
7784:
7774:
7759:
7746:
7741:
7738:
7733:
7729:
7726:
7720:
7717:
7712:
7709:
7693:
7692:
7689:
7686:
7679:
7676:
7673:
7672:
7661:
7658:
7655:
7652:
7648:
7645:
7642:
7639:
7634:
7630:
7626:
7623:
7620:
7616:
7613:
7610:
7607:
7602:
7598:
7587:
7584:
7568:
7565:
7562:
7560:
7558:
7555:
7551:
7548:
7545:
7542:
7539:
7536:
7533:
7529:
7526:
7523:
7520:
7517:
7516:
7513:
7510:
7507:
7505:
7503:
7500:
7497:
7494:
7491:
7485:
7482:
7477:
7474:
7468:
7465:
7462:
7459:
7456:
7455:
7435:
7434:
7423:
7420:
7414:
7411:
7408:
7405:
7400:
7397:
7389:
7385:
7381:
7378:
7368:
7358:
7342:
7339:
7335:
7332:
7329:
7326:
7323:
7320:
7318:
7316:
7313:
7310:
7309:
7306:
7303:
7300:
7297:
7294:
7291:
7289:
7284:
7281:
7276:
7273:
7267:
7266:
7250:
7249:
7238:
7235:
7232:
7229:
7225:
7222:
7219:
7216:
7211:
7207:
7203:
7200:
7190:
7187:
7171:
7168:
7164:
7161:
7158:
7155:
7152:
7149:
7147:
7145:
7142:
7139:
7138:
7135:
7132:
7129:
7126:
7123:
7120:
7118:
7113:
7110:
7105:
7102:
7096:
7095:
7079:
7078:
7067:
7064:
7061:
7058:
7051:
7048:
7045:
7040:
7036:
7030:
7027:
7024:
7019:
7015:
7006:
7002:
6998:
6995:
6992:
6985:
6982:
6979:
6974:
6970:
6964:
6961:
6958:
6953:
6949:
6940:
6936:
6925:
6921:
6915:
6909:
6893:
6890:
6887:
6885:
6883:
6880:
6876:
6873:
6870:
6865:
6861:
6857:
6854:
6851:
6846:
6842:
6838:
6835:
6832:
6828:
6825:
6822:
6817:
6813:
6809:
6806:
6803:
6798:
6794:
6790:
6789:
6786:
6783:
6780:
6778:
6773:
6770:
6765:
6762:
6755:
6752:
6749:
6744:
6740:
6736:
6733:
6730:
6725:
6721:
6717:
6714:
6711:
6708:
6703:
6699:
6695:
6692:
6689:
6684:
6680:
6676:
6675:
6654:
6653:
6650:
6647:
6640:
6637:
6575:
6568:
6455:
6452:
6451:
6450:
6439:
6436:
6433:
6428:
6424:
6420:
6417:
6414:
6409:
6405:
6401:
6398:
6395:
6392:
6389:
6384:
6380:
6376:
6373:
6370:
6365:
6361:
6357:
6354:
6351:
6346:
6342:
6338:
6335:
6332:
6327:
6323:
6319:
6316:
6313:
6309:
6304:
6301:
6296:
6292:
6288:
6285:
6282:
6277:
6273:
6269:
6266:
6262:
6247:
6246:
6235:
6231:
6227:
6224:
6221:
6217:
6213:
6209:
6205:
6189:
6188:
6173:
6170:
6165:
6161:
6157:
6154:
6151:
6146:
6142:
6138:
6135:
6132:
6129:
6126:
6124:
6121:
6118:
6114:
6110:
6106:
6105:
6100:
6096:
6092:
6090:
6087:
6084:
6080:
6077:
6074:
6070:
6066:
6065:
6061:
6058:
6056:
6051:
6047:
6043:
6041:
6038:
6035:
6031:
6027:
6023:
6022:
6017:
6013:
6009:
6007:
6004:
6001:
5997:
5993:
5989:
5988:
5962:
5961:
5949:
5944:
5941:
5938:
5935:
5932:
5928:
5924:
5919:
5915:
5896:
5895:
5882:
5879:
5876:
5872:
5868:
5864:
5858:
5855:
5852:
5849:
5846:
5842:
5835:
5832:
5826:
5822:
5819:
5815:
5811:
5808:
5804:
5801:
5798:
5795:
5791:
5787:
5768:
5765:
5756:
5753:
5749:
5748:
5737:
5734:
5731:
5728:
5725:
5722:
5719:
5716:
5713:
5706:
5702:
5698:
5695:
5691:
5666:
5641:
5638:
5633:
5629:
5624:
5620:
5617:
5612:
5608:
5604:
5601:
5597:
5581:
5580:
5567:
5562:
5559:
5554:
5550:
5546:
5543:
5539:
5536:
5533:
5526:
5522:
5518:
5513:
5508:
5504:
5499:
5495:
5494:
5491:
5488:
5483:
5479:
5475:
5472:
5464:
5460:
5456:
5451:
5446:
5442:
5438:
5435:
5432:
5428:
5424:
5423:
5420:
5417:
5412:
5408:
5404:
5401:
5397:
5396:
5394:
5378:
5377:
5363:
5360:
5355:
5351:
5347:
5344:
5341:
5336:
5332:
5328:
5322:
5318:
5312:
5309:
5306:
5303:
5300:
5276:
5245:
5244:
5231:
5227:
5223:
5219:
5216:
5204:
5203:
5199:
5198:
5194:
5187:
5181:
5169:
5157:
5135:
5132:
5126:
5111:
5110:
5096:
5093:
5087:
5084:
5078:
5075:
5072:
5069:
5062:
5058:
5054:
5051:
5047:
5035:
5024:
5021:
5018:
5015:
5012:
5009:
5006:
5003:
4996:
4992:
4988:
4985:
4981:
4980:lim sup
4956:
4953:
4950:
4945:
4941:
4915:
4911:
4895:
4894:
4881:
4877:
4873:
4868:
4864:
4860:
4857:
4854:
4851:
4848:
4845:
4841:
4837:
4832:
4828:
4824:
4821:
4816:
4812:
4808:
4803:
4799:
4795:
4792:
4787:
4783:
4767:
4760:
4750:
4747:
4704:
4701:
4698:
4693:
4689:
4685:
4682:
4679:
4674:
4670:
4666:
4663:
4660:
4657:
4654:
4649:
4645:
4641:
4638:
4635:
4630:
4626:
4622:
4619:
4616:
4554:
4551:
4548:
4543:
4539:
4535:
4532:
4529:
4524:
4520:
4516:
4513:
4510:
4507:
4504:
4499:
4495:
4491:
4488:
4485:
4480:
4476:
4472:
4469:
4466:
4434:
4429:
4425:
4421:
4418:
4415:
4410:
4406:
4401:
4397:
4394:
4391:
4388:
4385:
4382:
4379:
4375:
4372:
4359:
4356:
4341:
4340:
4337:
4336:
4333:
4327:
4321:
4320:
4317:
4311:
4305:
4304:
4301:
4298:
4282:
4279:
4263:eigenfunctions
4251:Main article:
4248:
4245:
4220:). Continuous
4186:
4183:
4155:Main article:
4152:
4149:
4123:
4120:
4097:The theory of
4094:
4091:
4089:
4086:
4085:
4084:
4071:
4066:
4061:
4055:
4052:
4047:
4044:
4040:
4036:
4031:
4028:
4023:
4020:
4016:
4009:
4006:
4001:
3998:
3995:
3992:
3989:
3975:
3974:
3963:
3960:
3957:
3954:
3951:
3948:
3943:
3937:
3933:
3929:
3923:
3920:
3917:
3909:
3906:
3902:
3899:
3879:
3876:
3802:
3801:
3789:
3786:
3783:
3780:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3680:
3679:
3668:
3665:
3662:
3659:
3655:
3652:
3649:
3644:
3641:
3638:
3634:
3627:
3624:
3618:
3614:
3611:
3607:
3604:
3601:
3598:
3595:
3592:
3562:integral curve
3532:
3531:
3520:
3517:
3513:
3507:
3504:
3501:
3497:
3493:
3490:
3487:
3483:
3480:
3476:
3473:
3470:
3467:
3463:
3459:
3444:
3441:
3437:phase portrait
3426:Hessian matrix
3400:
3396:
3391:
3387:
3383:
3379:
3375:
3372:
3369:
3365:
3361:
3334:
3330:
3326:
3321:
3317:
3312:
3308:
3304:
3301:
3297:
3292:
3280:
3279:
3266:
3260:
3257:
3256:
3253:
3250:
3249:
3246:
3243:
3242:
3239:
3236:
3235:
3233:
3228:
3223:
3217:
3212:
3209:
3206:
3201:
3196:
3193:
3190:
3186:
3182:
3177:
3173:
3169:
3164:
3160:
3156:
3153:
3150:
3145:
3141:
3137:
3136:
3133:
3130:
3129:
3126:
3121:
3118:
3115:
3110:
3105:
3102:
3099:
3095:
3091:
3086:
3082:
3078:
3073:
3069:
3065:
3062:
3059:
3054:
3050:
3046:
3045:
3042:
3037:
3034:
3031:
3026:
3021:
3018:
3015:
3011:
3007:
3002:
2998:
2994:
2989:
2985:
2981:
2978:
2975:
2970:
2966:
2962:
2961:
2959:
2935:
2934:
2922:
2918:
2914:
2908:
2905:
2902:
2897:
2892:
2889:
2886:
2882:
2878:
2873:
2869:
2865:
2860:
2856:
2852:
2849:
2845:
2840:
2821:
2820:
2807:
2800:
2794:
2791:
2788:
2785:
2782:
2777:
2772:
2769:
2766:
2762:
2758:
2753:
2749:
2745:
2740:
2736:
2732:
2729:
2725:
2719:
2715:
2711:
2710:
2707:
2704:
2703:
2699:
2693:
2690:
2687:
2684:
2681:
2676:
2671:
2668:
2665:
2661:
2657:
2652:
2648:
2644:
2639:
2635:
2631:
2628:
2624:
2618:
2614:
2610:
2609:
2605:
2599:
2596:
2593:
2590:
2587:
2582:
2577:
2574:
2571:
2567:
2563:
2558:
2554:
2550:
2545:
2541:
2537:
2534:
2530:
2524:
2520:
2516:
2515:
2513:
2508:
2503:
2495:
2492:
2489:
2484:
2480:
2476:
2475:
2472:
2469:
2468:
2463:
2460:
2457:
2452:
2448:
2444:
2443:
2438:
2435:
2432:
2427:
2423:
2419:
2418:
2416:
2378:
2377:
2365:
2359:
2356:
2353:
2350:
2347:
2342:
2337:
2334:
2331:
2327:
2323:
2318:
2314:
2310:
2305:
2301:
2297:
2294:
2290:
2285:
2281:
2276:
2273:
2270:
2265:
2221:Main article:
2218:
2217:System of ODEs
2215:
2214:
2213:
2210:
2205:
2203:
2202:
2188:nonhomogeneous
2184:
2179:
2177:
2148:
2143:
2140:
2098:
2091:
2090:
2079:
2076:
2073:
2070:
2067:
2062:
2059:
2056:
2052:
2048:
2045:
2042:
2037:
2033:
2027:
2024:
2021:
2016:
2013:
2010:
2006:
2002:
1997:
1994:
1991:
1987:
1954:
1939:
1934:
1932:
1919:
1914:
1908:
1907:
1896:
1893:
1889:
1883:
1880:
1877:
1873:
1866:
1863:
1857:
1853:
1850:
1846:
1842:
1839:
1835:
1832:
1829:
1826:
1822:
1818:
1786:
1785:
1772:
1769:
1766:
1762:
1758:
1754:
1748:
1745:
1742:
1739:
1736:
1732:
1728:
1725:
1722:
1718:
1715:
1711:
1708:
1705:
1702:
1698:
1694:
1657:
1654:
1641:
1635:
1632:
1629:
1625:
1620:
1614:
1611:
1605:
1599:
1596:
1590:
1441:
1438:
1400:. In general,
1390:
1389:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1348:
1344:
1339:
1333:
1330:
1327:
1324:
1319:
1314:
1306:
1182:
1179:
1112:
1109:
1106:
1102:
1098:
1095:
1092:
1088:
1085:
1069:are arbitrary
1056:
1053:
1050:
1047:
1023:
1020:
1017:
1012:
1008:
983:
980:
977:
972:
968:
954:
953:
942:
939:
936:
933:
930:
927:
924:
921:
916:
913:
910:
906:
902:
899:
896:
891:
887:
883:
880:
877:
873:
870:
866:
863:
860:
855:
851:
847:
843:
840:
836:
833:
830:
825:
821:
817:
814:
811:
808:
805:
800:
796:
768:
765:
755:
717:
716:
714:
713:
706:
699:
691:
688:
687:
683:
682:
677:
672:
667:
665:Ernst Lindelöf
662:
657:
652:
647:
642:
637:
635:Joseph Fourier
632:
627:
622:
620:Leonhard Euler
617:
612:
607:
601:
600:
597:
596:
593:
592:
588:
587:
584:
583:
578:
573:
566:
565:
560:
555:
550:
545:
540:
539:
538:
528:
523:
522:
521:
514:Finite element
511:
507:Crank–Nicolson
498:
493:
487:
482:
478:
477:
474:
473:
470:
469:
465:
464:
459:
454:
446:
441:
428:
426:Phase portrait
423:
418:
417:
416:
414:Cauchy problem
411:
406:
401:
391:
385:
384:
382:General topics
381:
380:
377:
376:
373:
372:
367:
362:
357:
351:
348:
347:
344:
343:
339:
338:
334:
333:
328:
327:
326:
315:
314:
313:
304:
301:
300:
297:
296:
291:
290:
289:
288:
281:
280:
275:
269:
266:
265:
261:
260:
258:
257:
255:Nonhomogeneous
248:
243:
240:
234:
233:
232:
224:
223:
219:
218:
216:
215:
210:
205:
200:
195:
190:
185:
179:
174:
171:
170:
167:
166:
165:Classification
162:
161:
152:
151:
150:
149:
144:
136:
135:
129:
128:
127:
126:
121:
116:
108:
107:
101:
100:
99:
98:
93:
87:
82:
77:
69:
68:
66:
65:
60:
54:
49:
46:
45:
42:
41:
37:
36:
26:
9:
6:
4:
3:
2:
14366:
14355:
14352:
14350:
14347:
14346:
14344:
14331:
14330:
14324:
14318:
14315:
14313:
14310:
14308:
14305:
14303:
14300:
14298:
14295:
14293:
14290:
14289:
14286:
14283:
14281:
14278:
14275:
14271:
14268:
14266:
14263:
14261:
14258:
14256:
14253:
14251:
14248:
14245:
14241:
14238:
14236:
14233:
14231:
14230:Real analysis
14228:
14227:
14224:
14218:
14215:
14213:
14210:
14208:
14205:
14203:
14200:
14198:
14195:
14193:
14190:
14188:
14185:
14181:
14178:
14176:
14173:
14171:
14168:
14167:
14166:
14163:
14161:
14158:
14156:
14152:
14151:
14147:
14146:
14143:
14139:
14131:
14126:
14124:
14119:
14117:
14112:
14111:
14108:
14096:
14093:
14091:
14088:
14086:
14083:
14081:
14078:
14076:
14073:
14071:
14068:
14066:
14063:
14061:
14058:
14056:
14053:
14051:
14048:
14046:
14043:
14041:
14038:
14036:
14033:
14031:
14028:
14026:
14023:
14022:
14020:
14016:
14010:
14007:
14005:
14002:
14000:
13997:
13995:
13992:
13991:
13989:
13985:
13975:
13972:
13970:
13967:
13965:
13962:
13960:
13957:
13955:
13952:
13950:
13947:
13945:
13942:
13940:
13937:
13935:
13932:
13930:
13927:
13925:
13922:
13920:
13917:
13915:
13912:
13910:
13907:
13905:
13902:
13901:
13899:
13895:
13889:
13886:
13884:
13881:
13878:
13875:
13873:
13870:
13868:
13865:
13863:
13860:
13858:
13855:
13853:
13850:
13848:
13845:
13843:
13840:
13838:
13835:
13834:
13832:
13828:
13822:
13819:
13817:
13814:
13812:
13809:
13807:
13804:
13803:
13801:
13797:
13794:
13790:
13780:
13777:
13773:
13770:
13769:
13768:
13765:
13762:
13759:
13758:
13756:
13752:
13746:
13743:
13741:
13738:
13736:
13733:
13731:
13728:
13726:
13723:
13721:
13718:
13716:
13713:
13711:
13708:
13706:
13703:
13701:
13698:
13697:
13695:
13691:
13685:
13682:
13680:
13677:
13675:
13672:
13670:
13667:
13665:
13662:
13660:
13657:
13655:
13652:
13650:
13647:
13645:
13642:
13640:
13637:
13636:
13634:
13630:
13627:
13623:
13619:
13612:
13607:
13605:
13600:
13598:
13593:
13592:
13589:
13583:
13580:
13577:
13574:
13571:
13567:
13564:
13561:
13560:Gerald Teschl
13557:
13554:
13551:
13548:
13545:
13542:
13539:
13535:
13532:
13529:
13526:
13522:
13518:
13517:
13512:
13508:
13507:
13498:
13494:
13492:
13491:0-415-27267-X
13488:
13484:
13480:
13477:
13473:
13467:
13463:
13459:
13455:
13454:
13449:
13445:
13440:
13438:0-8493-4488-3
13434:
13430:
13426:
13422:
13420:
13419:0-486-49510-8
13416:
13412:
13408:
13405:
13402:
13398:
13394:
13388:
13384:
13383:
13378:
13374:
13372:
13368:
13367:
13362:
13359:
13355:
13351:
13345:
13341:
13337:
13333:
13329:
13328:
13323:
13319:
13315:
13310:
13309:
13303:
13298:
13297:
13288:
13282:
13278:
13273:
13270:
13264:
13260:
13256:
13251:
13244:
13243:
13237:
13234:
13232:0-87901-432-6
13228:
13224:
13220:
13215:
13212:
13208:
13204:
13200:
13196:
13192:
13190:
13189:1-58488-297-2
13186:
13182:
13178:
13174:
13172:0-471-50728-8
13168:
13164:
13159:
13158:
13152:
13148:
13145:
13143:0-13-487538-9
13139:
13135:
13134:Prentice-Hall
13131:
13126:
13123:
13121:0-471-71716-9
13117:
13113:
13109:
13105:
13101:
13097:
13096:
13083:
13081:
13074:
13073:0-7135-1594-5
13070:
13064:
13055:
13053:
13043:
13037:
13036:0-471-83824-1
13033:
13027:
13025:
13023:
13021:
13019:
13017:
13015:
13013:
13011:
13009:
13000:
12995:, p. 13)
12994:
12989:
12980:
12972:
12966:
12961:
12954:
12949:
12941:
12937:
12933:
12929:
12925:
12921:
12914:
12907:
12902:
12896:, p. 78)
12895:
12890:
12882:
12876:
12871:
12863:
12857:
12853:
12846:
12838:
12833:, p. 12)
12832:
12827:
12821:, p. 12)
12820:
12815:
12808:
12803:
12797:, p. 47)
12796:
12795:Simmons (1972
12791:
12785:, p. 24)
12784:
12779:
12777:
12769:
12768:Simmons (1972
12764:
12757:
12752:
12745:
12740:
12738:
12730:
12725:
12719:, p. 78)
12718:
12713:
12706:
12705:Simmons (1972
12701:
12695:, p. 64)
12694:
12689:
12682:
12676:
12667:
12662:
12655:
12647:
12641:
12637:
12630:
12622:
12618:
12613:
12608:
12604:
12600:
12596:
12592:
12588:
12581:
12572:
12567:
12560:
12546:
12542:
12538:
12532:
12516:
12512:
12506:
12502:
12501:
12493:
12489:
12480:
12477:
12475:
12472:
12470:
12467:
12465:
12462:
12460:
12457:
12455:
12452:
12450:
12447:
12446:
12437:
12434:
12431:
12427:
12424:
12421:
12418:
12415:
12412:
12409:
12406:
12403:
12400:
12397:
12394:
12391:
12388:
12385:
12382:
12379:
12376:
12373:
12370:
12367:
12363:
12360:
12357:
12353:
12350:
12349:
12343:
12325:
12317:
12304:
12300:
12284:
12281:
12277:
12273:
12270:
12267:
12251:
12248:
12240:
12230:
12226:
12222:
12216:
12215:
12211:
12206:This section
12204:
12200:
12195:
12194:
12190:
12176:
12173:
12169:
12148:
12144:
12140:
12137:
12132:
12128:
12121:
12118:
12113:
12108:
12104:
12099:
12093:
12089:
12085:
12080:
12075:
12071:
12066:
12060:
12056:
12047:
12042:
12038:
12033:
12029:
12025:
12020:
12016:
11998:
11993:
11989:
11984:
11979:
11973:
11970:
11967:
11963:
11957:
11954:
11951:
11947:
11939:
11935:
11929:
11926:
11923:
11919:
11914:
11908:
11903:
11900:
11897:
11893:
11889:
11884:
11880:
11871:
11863:
11859:
11841:
11836:
11832:
11827:
11821:
11817:
11811:
11806:
11803:
11800:
11796:
11792:
11787:
11783:
11773:
11769:
11753:
11750:
11742:
11738:
11734:
11731:
11723:
11718:
11715:
11712:
11708:
11699:
11696:
11692:
11687:
11683:
11664:
11660:
11656:
11651:
11647:
11643:
11640:
11633:
11631:
11629:
11625:
11621:
11617:
11594:
11589:
11585:
11580:
11571:
11565:
11561:
11553:
11546:
11544:
11528:
11522:
11519:
11511:
11507:
11503:
11498:
11493:
11489:
11480:
11476:
11470:
11465:
11462:
11459:
11455:
11445:
11442:
11441:
11438:
11424:
11419:
11413:
11407:
11403:
11399:
11396:
11393:
11386:
11382:
11378:
11375:
11370:
11366:
11362:
11358:
11352:
11346:
11342:
11338:
11335:
11332:
11325:
11321:
11317:
11314:
11309:
11305:
11300:
11292:
11288:
11285:
11279:
11275:
11271:
11266:
11262:
11253:
11250:
11246:
11240:
11223:
11219:
11216:
11210:
11206:
11197:
11193:
11189:
11186:
11181:
11177:
11170:
11165:
11161:
11152:
11149:
11145:
11139:
11123:
11117:
11114:
11111:
11106:
11102:
11096:
11093:
11089:
11082:
11079:
11074:
11070:
11064:
11060:
11056:
11050:
11044:
11041:
11038:
11033:
11029:
11023:
11020:
11016:
11009:
11006:
11001:
10997:
10991:
10987:
10983:
10978:
10974:
10965:
10962:
10958:
10938:
10934:
10930:
10925:
10921:
10917:
10914:
10907:
10905:
10903:
10899:
10895:
10891:
10868:
10863:
10859:
10854:
10845:
10839:
10835:
10827:
10820:
10818:
10802:
10796:
10793:
10790:
10787:
10784:
10778:
10775:
10770:
10767:
10761:
10758:
10750:
10746:
10742:
10737:
10732:
10728:
10715:
10714:
10698:
10692:
10688:
10684:
10681:
10676:
10672:
10668:
10665:
10662:
10658:
10654:
10651:
10644:
10638:
10633:
10630:
10623:
10617:
10612:
10608:
10603:
10597:
10593:
10588:
10582:
10578:
10573:
10567:
10564:
10557:
10551:
10546:
10542:
10538:
10534:
10530:
10527:
10520:
10504:
10501:
10494:
10488:
10483:
10479:
10474:
10465:
10463:
10447:
10441:
10438:
10435:
10431:
10424:
10417:
10414:
10410:
10405:
10397:
10391:
10387:
10383:
10377:
10374:
10369:
10366:
10357:
10351:
10348:
10345:
10337:
10333:
10329:
10324:
10319:
10315:
10302:
10301:
10298:
10284:
10280:
10277:
10274:
10271:
10264:
10258:
10253:
10250:
10243:
10237:
10232:
10228:
10223:
10217:
10213:
10208:
10202:
10199:
10192:
10186:
10181:
10177:
10173:
10169:
10165:
10162:
10153:
10139:
10134:
10131:
10124:
10118:
10113:
10109:
10104:
10095:
10093:
10077:
10071:
10068:
10065:
10059:
10053:
10050:
10044:
10041:
10036:
10033:
10021:
10020:
10009:
10003:
9979:
9975:
9971:
9963:
9959:
9955:
9952:
9949:
9942:
9936:
9931:
9927:
9923:
9918:
9915:
9907:
9903:
9899:
9896:
9893:
9886:
9867:
9862:
9856:
9853:
9848:
9845:
9839:
9831:
9828:
9824:
9819:
9811:
9807:
9803:
9798:
9793:
9789:
9779:
9776:
9771:
9768:
9762:
9754:, substitute
9752:
9748:
9742:
9740:
9724:
9718:
9715:
9707:
9703:
9699:
9694:
9689:
9685:
9672:
9671:
9660:
9648:
9635:
9632:
9625:
9622:
9619:
9613:
9607:
9604:
9601:
9595:
9590:
9586:
9579:
9567:
9561:
9558:
9555:
9549:
9546:
9540:
9534:
9514:
9511:
9504:
9501:
9498:
9492:
9486:
9483:
9480:
9474:
9469:
9465:
9458:
9446:
9440:
9437:
9434:
9428:
9425:
9419:
9413:
9404:
9387:
9384:
9381:
9378:
9371:
9365:
9360:
9356:
9352:
9349:
9346:
9339:
9336:
9333:
9327:
9321:
9318:
9315:
9309:
9304:
9300:
9297:
9292:
9285:
9282:
9275:
9269:
9264:
9260:
9256:
9253:
9250:
9243:
9240:
9237:
9231:
9225:
9222:
9219:
9213:
9208:
9204:
9201:
9196:
9190:
9187:
9184:
9178:
9163:
9159:
9155:
9150:
9148:
9132:
9121:
9118:
9106:
9100:
9089:
9086:
9066:
9062:
9058:
9054:
9051:
9049:
9033:
9025:
9016:
9010:
9002:
8987:
8970:
8967:
8965:
8960:
8957:
8950:
8947:
8944:
8938:
8935:
8932:
8929:
8922:
8919:
8916:
8910:
8903:
8900:
8898:
8890:
8887:
8884:
8878:
8875:
8869:
8866:
8861:
8858:
8849:
8846:
8843:
8837:
8824:
8821:
8820:
8817:
8804:
8801:
8794:
8791:
8788:
8782:
8777:
8773:
8766:
8754:
8748:
8745:
8742:
8736:
8733:
8727:
8721:
8701:
8698:
8691:
8688:
8685:
8679:
8674:
8670:
8663:
8651:
8645:
8642:
8639:
8633:
8630:
8624:
8618:
8593:
8590:
8587:
8584:
8577:
8571:
8566:
8562:
8558:
8555:
8552:
8545:
8542:
8539:
8533:
8528:
8524:
8520:
8518:
8510:
8507:
8500:
8494:
8489:
8485:
8481:
8478:
8475:
8468:
8465:
8462:
8456:
8451:
8447:
8443:
8441:
8433:
8430:
8427:
8421:
8410:
8407:
8405:
8389:
8381:
8372:
8366:
8358:
8343:
8326:
8323:
8321:
8316:
8313:
8306:
8303:
8300:
8294:
8291:
8288:
8285:
8278:
8275:
8272:
8266:
8259:
8256:
8254:
8246:
8243:
8240:
8234:
8231:
8225:
8222:
8217:
8214:
8205:
8202:
8199:
8193:
8180:
8177:
8176:
8173:
8171:
8167:
8163:
8159:
8154:
8132:
8126:
8123:
8117:
8111:
8105:
8100:
8097:
8090:
8084:
8076:
8073:
8069:
8065:
8059:
8056:
8050:
8047:
8039:
8036:
8032:
8030:
8013:
8010:
8008:
8003:
8000:
7993:
7990:
7984:
7981:
7978:
7975:
7972:
7965:
7962:
7956:
7953:
7946:
7943:
7941:
7933:
7930:
7925:
7922:
7912:
7909:
7903:
7900:
7897:
7891:
7888:
7882:
7879:
7866:
7865:
7847:
7844:
7838:
7832:
7827:
7824:
7816:
7812:
7808:
7804:
7800:
7794:
7791:
7785:
7782:
7775:
7772:
7768:
7764:
7760:
7758:
7744:
7739:
7736:
7731:
7727:
7724:
7718:
7715:
7710:
7707:
7695:
7694:
7683:
7659:
7656:
7653:
7650:
7643:
7637:
7632:
7628:
7624:
7621:
7618:
7611:
7605:
7600:
7596:
7588:
7585:
7583:
7566:
7563:
7561:
7556:
7553:
7546:
7540:
7537:
7534:
7531:
7524:
7518:
7511:
7508:
7506:
7498:
7492:
7489:
7483:
7480:
7475:
7472:
7463:
7457:
7445:
7441:
7437:
7436:
7421:
7418:
7409:
7403:
7398:
7395:
7387:
7383:
7379:
7376:
7369:
7366:
7362:
7359:
7357:
7340:
7337:
7330:
7324:
7321:
7319:
7314:
7311:
7301:
7295:
7292:
7290:
7282:
7279:
7274:
7271:
7256:
7252:
7251:
7236:
7233:
7230:
7227:
7220:
7214:
7209:
7205:
7201:
7198:
7191:
7188:
7186:
7169:
7166:
7159:
7153:
7150:
7148:
7143:
7140:
7130:
7124:
7121:
7119:
7111:
7108:
7103:
7100:
7085:
7081:
7080:
7065:
7062:
7059:
7056:
7046:
7038:
7034:
7025:
7017:
7013:
7004:
7000:
6996:
6993:
6990:
6980:
6972:
6968:
6959:
6951:
6947:
6938:
6934:
6926:
6920:
6914:
6910:
6908:
6891:
6888:
6886:
6881:
6878:
6871:
6863:
6859:
6852:
6844:
6840:
6836:
6833:
6830:
6823:
6815:
6811:
6804:
6796:
6792:
6784:
6781:
6779:
6771:
6768:
6763:
6760:
6750:
6742:
6738:
6731:
6723:
6719:
6715:
6709:
6701:
6697:
6690:
6682:
6678:
6664:
6660:
6656:
6655:
6644:
6636:
6633:
6629:
6624:
6614:
6610:
6604:
6600:
6596:
6590:
6585:
6583:
6574:
6567:
6561:
6555:
6549:
6543:
6539:functions of
6538:
6532:
6528:
6524:
6517:
6513:
6509:
6502:
6498:
6491:
6487:
6480:
6476:
6469:
6465:
6459:
6437:
6426:
6422:
6418:
6415:
6412:
6407:
6403:
6399:
6396:
6390:
6387:
6382:
6378:
6374:
6371:
6368:
6363:
6359:
6352:
6344:
6340:
6336:
6333:
6330:
6325:
6321:
6317:
6314:
6302:
6294:
6290:
6286:
6283:
6280:
6275:
6271:
6264:
6252:
6251:
6250:
6225:
6222:
6211:
6207:
6194:
6193:
6192:
6171:
6163:
6159:
6155:
6152:
6149:
6144:
6140:
6136:
6133:
6127:
6122:
6116:
6112:
6108:
6098:
6094:
6088:
6082:
6078:
6075:
6072:
6068:
6059:
6049:
6045:
6039:
6033:
6029:
6025:
6015:
6011:
6005:
5999:
5995:
5991:
5979:
5978:
5977:
5975:
5971:
5968:= 1, 2, ...,
5967:
5947:
5939:
5936:
5933:
5926:
5922:
5917:
5913:
5905:
5904:
5903:
5901:
5877:
5870:
5866:
5862:
5853:
5850:
5847:
5840:
5833:
5830:
5824:
5820:
5817:
5813:
5809:
5806:
5802:
5799:
5796:
5793:
5789:
5785:
5778:
5777:
5776:
5774:
5764:
5762:
5752:
5735:
5720:
5714:
5704:
5700:
5693:
5681:
5680:
5679:
5654:
5639:
5631:
5627:
5622:
5618:
5615:
5610:
5606:
5586:
5560:
5557:
5552:
5548:
5541:
5534:
5531:
5524:
5520:
5516:
5511:
5506:
5502:
5497:
5489:
5486:
5481:
5477:
5470:
5462:
5458:
5454:
5449:
5444:
5440:
5436:
5430:
5426:
5418:
5415:
5410:
5406:
5392:
5383:
5382:
5381:
5361:
5358:
5353:
5349:
5342:
5339:
5334:
5330:
5320:
5316:
5310:
5304:
5298:
5291:
5290:
5289:
5264:
5262:
5258:
5254:
5250:
5229:
5225:
5221:
5217:
5214:
5206:
5205:
5201:
5200:
5193:
5186:
5182:
5158:
5155:
5154:
5153:
5150:
5116:
5082:
5073:
5067:
5060:
5056:
5049:
5036:
5010:
5004:
4994:
4990:
4983:
4970:
4969:
4968:
4951:
4948:
4943:
4939:
4929:
4909:
4900:
4875:
4871:
4866:
4862:
4858:
4849:
4843:
4835:
4830:
4826:
4822:
4814:
4810:
4806:
4801:
4797:
4790:
4781:
4773:
4772:
4771:
4766:
4759:
4754:
4746:
4744:
4740:
4736:
4732:
4728:
4723:
4719:
4699:
4696:
4691:
4687:
4683:
4680:
4677:
4672:
4668:
4661:
4655:
4652:
4647:
4643:
4639:
4636:
4633:
4628:
4624:
4617:
4614:
4606:
4602:
4593:
4589:
4585:
4580:
4576:
4572:
4569:plane, where
4568:
4549:
4546:
4541:
4537:
4533:
4530:
4527:
4522:
4518:
4511:
4505:
4502:
4497:
4493:
4489:
4486:
4483:
4478:
4474:
4467:
4464:
4456:
4452:
4448:
4427:
4423:
4416:
4413:
4408:
4404:
4399:
4392:
4389:
4386:
4380:
4377:
4373:
4370:
4355:
4353:
4348:
4346:
4334:
4332:
4328:
4326:
4323:
4322:
4318:
4316:
4312:
4310:
4307:
4306:
4302:
4299:
4296:
4295:
4292:
4291:
4290:
4288:
4278:
4276:
4272:
4268:
4264:
4260:
4254:
4244:
4241:
4239:
4235:
4231:
4227:
4223:
4219:
4215:
4210:
4206:
4204:
4200:
4196:
4192:
4182:
4180:
4176:
4172:
4168:
4164:
4158:
4148:
4145:
4141:
4137:
4133:
4129:
4119:
4117:
4113:
4109:
4104:
4100:
4069:
4064:
4059:
4053:
4050:
4045:
4042:
4038:
4034:
4029:
4026:
4021:
4018:
4014:
4007:
4004:
3999:
3993:
3987:
3980:
3979:
3978:
3961:
3958:
3952:
3946:
3941:
3931:
3918:
3907:
3904:
3900:
3897:
3889:
3888:
3887:
3884:
3875:
3873:
3869:
3865:
3861:
3857:
3852:
3850:
3846:
3842:
3838:
3834:
3830:
3826:
3822:
3817:
3815:
3811:
3807:
3787:
3784:
3781:
3778:
3771:
3765:
3762:
3756:
3750:
3743:
3742:
3741:
3738:
3734:
3729:
3725:
3722:is called an
3721:
3716:
3712:
3708:
3704:
3698:
3694:
3690:
3686:
3666:
3663:
3660:
3657:
3653:
3650:
3639:
3632:
3625:
3622:
3616:
3612:
3609:
3605:
3602:
3599:
3596:
3590:
3583:
3582:
3581:
3579:
3575:
3571:
3567:
3563:
3559:
3555:
3550:
3546:
3542:
3538:
3518:
3515:
3511:
3502:
3495:
3491:
3488:
3485:
3481:
3478:
3474:
3471:
3468:
3465:
3461:
3457:
3450:
3449:
3448:
3440:
3438:
3433:
3431:
3427:
3423:
3418:
3381:
3373:
3370:
3349:
3328:
3324:
3319:
3310:
3302:
3299:
3295:
3264:
3258:
3251:
3244:
3237:
3231:
3226:
3221:
3207:
3194:
3191:
3188:
3184:
3175:
3171:
3162:
3154:
3151:
3143:
3139:
3131:
3116:
3103:
3100:
3097:
3093:
3084:
3080:
3071:
3063:
3060:
3052:
3048:
3032:
3019:
3016:
3013:
3009:
3000:
2996:
2987:
2979:
2976:
2968:
2964:
2957:
2948:
2947:
2946:
2944:
2940:
2916:
2912:
2903:
2890:
2887:
2884:
2880:
2871:
2867:
2858:
2850:
2847:
2843:
2830:
2829:
2828:
2827:analogue is:
2826:
2805:
2798:
2789:
2786:
2783:
2770:
2767:
2764:
2760:
2751:
2747:
2738:
2730:
2727:
2723:
2717:
2713:
2705:
2697:
2688:
2685:
2682:
2669:
2666:
2663:
2659:
2650:
2646:
2637:
2629:
2626:
2622:
2616:
2612:
2603:
2594:
2591:
2588:
2575:
2572:
2569:
2565:
2556:
2552:
2543:
2535:
2532:
2528:
2522:
2518:
2511:
2506:
2501:
2490:
2482:
2478:
2470:
2458:
2450:
2446:
2433:
2425:
2421:
2414:
2405:
2404:
2403:
2401:
2400:column vector
2397:
2394:
2390:
2387:
2383:
2363:
2354:
2351:
2348:
2335:
2332:
2329:
2325:
2316:
2312:
2303:
2295:
2292:
2288:
2279:
2271:
2254:
2253:
2252:
2250:
2246:
2242:
2238:
2234:
2230:
2224:
2211:
2208:
2198:
2194:
2189:
2185:
2182:
2173:
2168:
2162:
2158:
2153:
2149:
2146:
2141:
2138:
2134:
2130:
2120:
2116:
2109:
2101:
2096:
2074:
2068:
2065:
2057:
2050:
2043:
2035:
2031:
2025:
2022:
2019:
2014:
2011:
2008:
2004:
2000:
1992:
1985:
1977:
1976:
1973:
1968:
1952:
1944:
1940:
1937:
1929:
1924:
1920:
1917:
1912:
1911:
1894:
1891:
1887:
1878:
1871:
1864:
1861:
1855:
1851:
1848:
1844:
1840:
1837:
1833:
1830:
1827:
1824:
1820:
1816:
1809:
1808:
1807:
1805:
1801:
1796:
1794:
1792:
1788:is called an
1767:
1760:
1756:
1752:
1743:
1740:
1737:
1730:
1726:
1723:
1720:
1716:
1713:
1709:
1706:
1703:
1700:
1696:
1692:
1685:
1684:
1683:
1681:
1677:
1673:
1669:
1663:
1653:
1633:
1630:
1627:
1623:
1618:
1612:
1609:
1603:
1597:
1594:
1581:
1577:
1571:
1567:
1563:
1558:
1554:
1544:
1536:
1533:
1521:
1513:
1510:
1498:
1490:
1481:
1477:
1473:
1470:
1464:
1460:
1456:
1451:
1447:
1437:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1395:
1369:
1363:
1357:
1354:
1346:
1342:
1328:
1322:
1317:
1304:
1297:
1296:
1295:
1293:
1289:
1286:and the time
1285:
1281:
1276:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1245:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1202:
1200:
1196:
1192:
1188:
1178:
1176:
1172:
1171:Taylor series
1168:
1163:
1161:
1157:
1153:
1149:
1145:
1141:
1136:
1107:
1100:
1096:
1093:
1090:
1086:
1083:
1072:
1051:
1045:
1018:
1010:
1006:
978:
970:
966:
940:
937:
934:
928:
922:
919:
911:
904:
897:
889:
885:
881:
878:
875:
871:
868:
861:
853:
849:
845:
841:
838:
831:
823:
819:
815:
812:
806:
798:
794:
786:
785:
784:
783:of the form
782:
778:
774:
764:
762:
760:
753:
751:
749:
744:
740:
736:
732:
728:
724:
712:
707:
705:
700:
698:
693:
692:
690:
689:
681:
678:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
616:
613:
611:
608:
606:
603:
602:
595:
594:
590:
589:
582:
579:
577:
574:
572:
569:
568:
564:
561:
559:
556:
554:
551:
549:
546:
544:
541:
537:
534:
533:
532:
529:
527:
526:Finite volume
524:
520:
517:
516:
515:
512:
508:
502:
499:
497:
494:
492:
488:
486:
483:
480:
479:
472:
471:
463:
460:
458:
455:
451:
447:
445:
442:
440:
436:
432:
429:
427:
424:
422:
419:
415:
412:
410:
407:
405:
402:
400:
397:
396:
395:
392:
390:
387:
386:
379:
378:
371:
368:
366:
363:
361:
358:
356:
353:
352:
346:
345:
341:
340:
332:
329:
325:
322:
321:
320:
317:
316:
312:
306:
305:
299:
298:
287:
284:
283:
279:
276:
274:
271:
270:
268:
267:
263:
262:
256:
252:
249:
247:
244:
241:
239:
236:
235:
231:
228:
227:
226:
225:
221:
220:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
186:
184:
181:
180:
178:
177:
169:
168:
164:
163:
158:
148:
145:
143:
140:
139:
138:
137:
134:
131:
130:
125:
122:
120:
117:
115:
112:
111:
110:
109:
106:
103:
102:
97:
94:
92:
88:
86:
83:
81:
78:
76:
73:
72:
71:
70:
64:
61:
59:
56:
55:
53:
52:
44:
43:
39:
38:
35:
32:
31:
19:
14327:
14169:
14148:
14090:Martin Kutta
14045:Émile Picard
14025:Isaac Newton
13939:Euler method
13909:Substitution
13648:
13514:
13496:
13482:
13452:
13428:
13410:
13381:
13365:
13363:W. Johnson,
13326:
13312:. New York:
13307:
13294:Bibliography
13276:
13254:
13241:
13218:
13201:, New York:
13198:
13180:
13156:
13129:
13107:
13063:
13042:
12993:Ascher (1998
12988:
12979:
12967:, p. 9)
12960:
12955:, 1866, 1868
12952:
12948:
12923:
12913:
12908:, p. 4)
12901:
12889:
12877:, p. 5)
12875:Ascher (1998
12870:
12851:
12845:
12831:Ascher (1998
12826:
12814:
12807:Harper (1976
12802:
12790:
12770:, p. 3)
12763:
12758:, p. 2)
12751:
12744:Harper (1976
12729:Tipler (1991
12724:
12712:
12700:
12688:
12675:
12654:
12635:
12629:
12594:
12590:
12580:
12559:
12548:. Retrieved
12540:
12531:
12519:. Retrieved
12499:
12492:
12305:
12301:
12258:
12243:
12237:January 2020
12234:
12219:Please help
12207:
12171:
12167:
12044:, and using
12040:
12036:
12031:
12027:
12023:
12018:
12014:
11866:
11861:
11857:
11771:
11767:
11766:, then: for
11697:
11685:
11681:
11679:
11627:
11623:
11612:
11610:
11563:
11556:
11548:
11447:
11443:
11254:
11248:
11244:
11241:
11153:
11147:
11143:
11140:
10966:
10960:
10956:
10953:
10901:
10897:
10886:
10884:
10837:
10830:
10822:
10717:
10304:
10155:
10023:
10001:
9750:
9746:
9674:
9405:
9167:
9161:
9157:
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9070:
9064:
9060:
9056:
8988:
8826:
8610:
8344:
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8165:
8161:
8157:
8155:
8040:
8034:
7868:
7770:
7766:
7762:
7697:
7446:
7443:
7439:
7364:
7257:
7254:
7086:
7083:
6918:
6912:
6666:
6662:
6658:
6631:
6627:
6622:
6612:
6608:
6602:
6598:
6594:
6586:
6572:
6565:
6559:
6553:
6547:
6541:
6530:
6526:
6522:
6515:
6511:
6507:
6500:
6496:
6489:
6485:
6478:
6474:
6467:
6463:
6460:
6457:
6248:
6190:
5973:
5969:
5965:
5963:
5899:
5897:
5772:
5770:
5758:
5750:
5655:
5584:
5582:
5379:
5265:
5260:
5256:
5252:
5248:
5246:
5191:
5184:
5151:
5114:
5112:
4930:
4896:
4764:
4757:
4755:
4752:
4742:
4738:
4734:
4730:
4726:
4721:
4717:
4599:denotes the
4591:
4587:
4583:
4574:
4570:
4566:
4454:
4450:
4446:
4361:
4349:
4342:
4284:
4256:
4242:
4226:Lie algebras
4222:group theory
4211:
4207:
4188:
4185:Lie's theory
4178:
4160:
4131:
4125:
4096:
3976:
3885:
3881:
3859:
3855:
3853:
3836:
3828:
3824:
3820:
3818:
3813:
3812:is called a
3809:
3805:
3803:
3736:
3732:
3727:
3723:
3719:
3714:
3710:
3706:
3702:
3696:
3692:
3688:
3684:
3681:
3577:
3573:
3569:
3565:
3557:
3553:
3548:
3544:
3540:
3536:
3533:
3446:
3434:
3417:non-singular
3281:
2938:
2936:
2824:
2822:
2395:
2392:
2388:
2385:
2381:
2379:
2248:
2240:
2236:
2232:
2228:
2226:
2196:
2192:
2187:
2171:
2160:
2156:
2151:
2136:
2132:
2128:
2118:
2114:
2107:
2099:
2094:
1971:
1942:
1927:
1922:
1909:
1803:
1799:
1797:
1789:
1787:
1679:
1675:
1671:
1667:
1665:
1569:
1565:
1561:
1542:
1534:
1531:
1519:
1511:
1508:
1496:
1488:
1471:
1462:
1458:
1454:
1445:
1443:
1433:
1429:
1425:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1391:
1291:
1287:
1283:
1277:
1246:
1203:
1184:
1164:
1137:
955:
770:
758:
747:
730:
726:
720:
670:Émile Picard
655:Martin Kutta
645:George Green
605:Isaac Newton
437: /
433: /
253: /
182:
119:Chaos theory
14155:Integration
13847:Phase space
13705:Homogeneous
13314:McGraw-Hill
13248:(in French)
13203:McGraw-Hill
12597:(1): 1–29.
12396:Mathematica
9069:satisfying
7363:(divide by
5259:, which is
4899:restriction
4303:Conclusion
4300:Assumption
4259:eigenvalues
4189:From 1870,
4128:quadratures
3860:homogeneous
3534:a function
2943:zero vector
2152:homogeneous
2145:Homogeneous
2137:source term
1553:integration
1440:Definitions
1222:meteorology
1187:mathematics
743:derivatives
723:mathematics
563:Runge–Kutta
308:Difference
251:Homogeneous
63:Engineering
14343:Categories
14180:stochastic
14075:John Crank
13904:Inspection
13767:Stochastic
13761:Difference
13735:Autonomous
13679:Non-linear
13669:Fractional
13632:Operations
13458:Providence
13093:References
12666:2012.07558
12571:2206.00364
12550:2016-07-28
12378:GNU Octave
12364:, a free (
11691:polynomial
11572:functions
10846:functions
6537:integrable
4315:continuous
4218:Lie theory
4195:Lie groups
4191:Sophus Lie
2239:) = , and
2207:Non-linear
1923:autonomous
1916:Autonomous
1660:See also:
1555:, whereas
1269:d'Alembert
1181:Background
1140:elementary
759:stochastic
680:John Crank
481:Inspection
435:Asymptotic
319:Stochastic
238:Autonomous
213:Non-linear
203:Fractional
14292:Functions
13879:solutions
13837:Wronskian
13792:Solutions
13720:Decoupled
13684:Holonomic
13521:EMS Press
13379:(1944) ,
13324:(2002) ,
12621:0377-0427
12282:α
12208:does not
12145:φ
12129:γ
12122:
12105:χ
12072:α
12013:for some
11990:α
11971:−
11968:ℓ
11958:ℓ
11924:ℓ
11920:∑
11894:∑
11865:repeated
11833:α
11797:∑
11739:α
11735:−
11732:α
11709:∏
11586:α
11554:: assume
11456:∑
11400:−
11379:
11339:−
11318:
11280:−
11211:−
11112:−
11097:−
11075:−
11039:−
11002:−
10860:α
10828:: assume
10666:ξ
10655:λ
10645:λ
10634:ε
10624:ε
10613:λ
10609:∫
10598:ξ
10594:∫
10579:∫
10568:λ
10558:λ
10543:∫
10539:−
10505:λ
10495:λ
10480:∫
10275:λ
10265:λ
10254:ε
10244:ε
10233:λ
10229:∫
10214:∫
10203:λ
10193:λ
10178:∫
10174:−
10135:λ
10125:λ
10110:∫
9953:ε
9943:ε
9932:λ
9928:∫
9919:λ
9904:∫
9900:±
9636:λ
9626:λ
9608:λ
9596:μ
9587:∫
9577:∂
9573:∂
9568:−
9515:λ
9499:λ
9481:λ
9475:μ
9466:∫
9456:∂
9452:∂
9447:−
9382:λ
9372:λ
9357:∫
9350:λ
9340:λ
9322:λ
9310:μ
9301:∫
9286:λ
9276:λ
9261:∫
9254:λ
9238:λ
9220:λ
9214:μ
9205:∫
9130:∂
9119:μ
9113:∂
9098:∂
9087:μ
9081:∂
9031:∂
9023:∂
9017:≠
9008:∂
9000:∂
8805:λ
8795:λ
8774:∫
8764:∂
8760:∂
8755:−
8702:λ
8686:λ
8671:∫
8661:∂
8657:∂
8652:−
8588:λ
8578:λ
8563:∫
8556:λ
8546:λ
8525:∫
8511:λ
8501:λ
8486:∫
8479:λ
8463:λ
8448:∫
8387:∂
8379:∂
8364:∂
8356:∂
8133:λ
8124:−
8118:λ
8106:λ
8101:λ
8091:λ
8070:∫
8051:
7848:λ
7845:−
7839:λ
7828:λ
7805:∫
7786:
7654:λ
7644:λ
7629:∫
7622:λ
7612:λ
7597:∫
7410:λ
7399:λ
7384:∫
7231:λ
7221:λ
7206:∫
7060:λ
7047:λ
7026:λ
7001:∫
6994:λ
6981:λ
6960:λ
6935:∫
6589:summation
6416:…
6372:…
6334:…
6284:…
6153:…
6076:−
6060:⋮
5937:−
5851:−
5831:…
5733:∞
5730:→
5727:‖
5712:‖
5705:±
5697:→
5600:∖
5538:∞
5434:∞
5431:−
5340:−
5134:¯
5131:Ω
5125:∂
5095:¯
5092:Ω
5086:∂
5083:∈
5061:±
5053:→
5023:∞
5020:→
5017:‖
5002:‖
4995:±
4987:→
4955:∞
4952:±
4949:≠
4944:±
4872:∈
4853:∞
4850:±
4844:∪
4836:∈
4831:±
4802:−
4715:for some
4678:−
4662:⊂
4634:−
4528:−
4512:×
4484:−
4347:are met.
4234:Lax pairs
4167:Frobenius
4046:−
4022:−
3908:−
3782:∈
3724:extension
3661:∈
3623:…
3489:…
3443:Solutions
3395:∂
3360:∂
3252:⋮
3192:…
3132:⋮
3101:…
3017:…
2888:…
2787:−
2768:…
2706:⋮
2686:−
2667:…
2592:−
2573:…
2471:⋮
2393:dimension
2352:−
2333:…
2023:−
2005:∑
1862:…
1741:−
1724:…
1613:¨
1598:˙
1242:economics
1226:chemistry
1218:astronomy
1199:gradients
1167:integrals
1094:…
879:⋯
754:more than
421:Wronskian
399:Dirichlet
142:Economics
85:Chemistry
75:Astronomy
14317:Infinity
14170:ordinary
14150:Calculus
13987:Examples
13877:Integral
13649:Ordinary
13450:(2012).
13427:(1993).
13304:(1955).
13279:, SIAM,
13211:75173716
13197:(1972),
13153:(1972),
13106:(1977),
12940:45426376
12515:Archived
12443:See also
12414:SageMath
10418:′
6535:are any
6208:′
6117:′
6083:′
6034:′
6000:′
5821:″
5810:′
5678:because
5218:′
5202:Example.
4374:′
4297:Theorem
4112:Casorati
4088:Theories
3901:′
3613:′
3558:solution
3552:, where
3482:′
3320:′
3185:″
3172:′
3094:″
3081:′
3010:″
2997:′
2881:″
2868:′
2825:implicit
2761:″
2748:′
2660:″
2647:′
2566:″
2553:′
2326:″
2313:′
2117: (
1852:″
1841:′
1800:implicit
1791:explicit
1717:′
1265:Clairaut
1206:geometry
1087:′
872:″
842:′
781:equation
739:variable
531:Galerkin
431:Lyapunov
342:Solution
286:Notation
278:Operator
264:Features
183:Ordinary
14175:partial
13715:Coupled
13654:Partial
13523:, 2001
13401:0010757
13358:1929104
13108:Physics
12599:Bibcode
12521:11 July
12426:Chebfun
12229:removed
12214:sources
11872:times,
11252:, then
11151:, then
10964:, then
6621:, then
6582:complex
4565:in the
4171:Clebsch
4108:Darboux
2104:
2097:
1568:′′, …,
1547:
1528:
1524:
1505:
1501:
1485:
1467:of the
1261:Riccati
1253:Leibniz
1234:ecology
1230:biology
1214:physics
1148:physics
1144:special
1125:
1075:
1067:
1038:
1034:
998:
996:, ...,
994:
958:
748:partial
733:) is a
404:Neumann
188:Partial
96:Geology
91:Biology
80:Physics
14312:Series
13730:Degree
13674:Linear
13489:
13468:
13435:
13417:
13399:
13389:
13356:
13346:
13283:
13265:
13229:
13209:
13187:
13169:
13140:
13118:
13071:
13034:
12953:Crelle
12938:
12858:
12642:
12619:
12507:
12430:MATLAB
12384:Scilab
12372:MATLAB
12362:COPASI
12352:Maxima
12166:where
11695:degree
11680:Since
11247:< 4
10959:> 4
9406:where
8989:where
8611:where
8345:where
7763:y = ux
6505:, and
6249:where
5972:. The
5837:
5828:
5080:
4725:where
4595:) and
4228:, and
4144:Cauchy
4116:Cayley
3847:'. A
3823:of an
3629:
3620:
3580:, and
2937:where
2402:form:
2380:is an
2092:where
1943:linear
1936:Linear
1868:
1859:
1666:Given
1474:. The
1271:, and
1255:, the
1249:Newton
1191:social
956:where
591:People
503:
450:Series
208:Linear
47:Fields
14307:Limit
13779:Delay
13725:Order
13246:(PDF)
13163:Wiley
13112:Wiley
12936:S2CID
12661:arXiv
12566:arXiv
12485:Notes
12436:GNU R
12420:SciPy
12402:SymPy
12390:Maple
6623:after
6578:, ...
5761:order
4449:and ∂
4163:Fuchs
4136:Gauss
3835:. A
3568:, if
2398:. In
2386:order
2243:is a
2199:) ≠ 0
2163:) = 0
1526:, …,
1448:is a
1273:Euler
725:, an
491:Euler
409:Robin
331:Delay
273:Order
246:Exact
172:Types
40:Scope
13570:UIUC
13487:ISBN
13466:ISBN
13433:ISBN
13415:ISBN
13387:ISBN
13344:ISBN
13281:ISBN
13263:ISBN
13227:ISBN
13207:LCCN
13185:ISBN
13167:ISBN
13138:ISBN
13116:ISBN
13069:ISBN
13032:ISBN
12999:help
12971:help
12881:help
12856:ISBN
12837:help
12640:ISBN
12617:ISSN
12523:2019
12505:ISBN
12212:any
12210:cite
9527:and
8714:and
7769:and
7761:Set
7442:and
6661:and
6557:and
5964:for
5558:<
5487:>
5255:) =
5253:x, y
4579:real
4577:are
4573:and
4273:and
4114:and
3870:and
3740:and
3700:and
3564:for
2391:and
2112:and
1436:)).
1236:and
1216:and
1208:and
1193:and
1189:and
1150:and
1142:and
1036:and
598:List
13336:doi
12928:doi
12607:doi
12595:125
12223:by
12119:cos
11693:of
11376:cos
11315:sin
11242:If
11146:= 4
11141:If
10954:If
9151:If
8156:If
8037:).
7367:).
6924:).
5690:lim
5046:lim
4914:max
4880:max
4786:max
4727:the
4567:x-y
4445:if
4352:DAE
3912:sgn
3843:or
3730:if
3726:of
3572:is
3560:or
3415:be
2384:of
2247:of
2190:if
2174:= 0
2154:if
1945:if
1564:′,
1263:,
1162:).
731:ODE
721:In
14345::
14153::
13519:,
13513:,
13464:.
13460::
13456:.
13409:,
13397:MR
13395:,
13354:MR
13352:,
13342:,
13334:,
13261:,
13225:,
13205:,
13165:,
13136:,
13114:,
13102:;
13079:^
13051:^
13007:^
12934:.
12922:.
12775:^
12736:^
12615:.
12605:.
12589:.
12543:.
12539:.
12513:.
12037:iγ
12035:+
12026:=
11700::
11609:.
11562:=
10883:.
10836:=
9751:dx
9747:dy
9160:,
9063:,
8172:.
8168:=
8166:xy
8160:=
8048:ln
8035:xy
7783:ln
7773:.
6630:=
6603:dλ
6601:)
6593:∫
6571:,
6551:;
6545:,
6520:,
6494:,
6483:,
6472:,
5775:,
5197:).
5190:,
4928:.
4763:,
4741:,
4720:∈
4590:∈
4586:,
4453:/∂
4329:F
4313:F
4224:,
4205:.
3874:.
3819:A
3816:.
3735:⊂
3718:,
3713:→
3709:⊂
3705::
3695:→
3691:⊂
3687::
3547:→
3543:⊂
3539::
3439:.
2201:.
2169:"
1795:.
1674:,
1543:dx
1520:dx
1503:,
1497:dx
1489:dy
1457:=
1275:.
1267:,
1259:,
1251:,
1135:.
771:A
14276:)
14272:(
14246:)
14242:(
14129:e
14122:t
14115:v
13610:e
13603:t
13596:v
13572:.
13562:.
13540:.
13474:.
13443:.
13441:.
13338::
13316:.
13176:.
13001:)
12973:)
12942:.
12930::
12883:)
12864:.
12839:)
12669:.
12663::
12648:.
12623:.
12609::
12601::
12574:.
12568::
12553:.
12525:.
12358:.
12326:+
12318:=
12285:t
12278:e
12274:A
12271:=
12268:y
12250:)
12244:(
12239:)
12235:(
12231:.
12217:.
12172:j
12168:ϕ
12154:)
12149:j
12141:+
12138:x
12133:j
12125:(
12114:x
12109:j
12100:e
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12090:C
12086:=
12081:x
12076:j
12067:e
12061:j
12057:C
12041:j
12032:j
12028:χ
12024:α
12019:j
12015:α
11999:x
11994:j
11985:e
11980:)
11974:1
11964:x
11955:,
11952:j
11948:C
11940:j
11936:k
11930:1
11927:=
11915:(
11909:n
11904:1
11901:=
11898:j
11890:=
11885:c
11881:y
11869:j
11867:k
11862:j
11858:α
11842:x
11837:j
11828:e
11822:j
11818:C
11812:n
11807:1
11804:=
11801:j
11793:=
11788:c
11784:y
11772:j
11768:α
11754:0
11751:=
11748:)
11743:j
11729:(
11724:n
11719:1
11716:=
11713:j
11698:n
11686:j
11682:α
11665:p
11661:y
11657:+
11652:c
11648:y
11644:=
11641:y
11628:x
11626:(
11624:r
11615:p
11613:y
11595:x
11590:j
11581:e
11566:,
11564:e
11559:c
11557:y
11551:c
11549:y
11532:)
11529:x
11526:(
11523:r
11520:=
11512:j
11508:x
11504:d
11499:y
11494:j
11490:d
11481:j
11477:b
11471:n
11466:0
11463:=
11460:j
11444:n
11425:]
11420:)
11414:2
11408:2
11404:b
11397:c
11394:4
11387:x
11383:(
11371:2
11367:C
11363:+
11359:)
11353:2
11347:2
11343:b
11336:c
11333:4
11326:x
11322:(
11310:1
11306:C
11301:[
11293:2
11289:x
11286:b
11276:e
11272:=
11267:c
11263:y
11249:c
11245:b
11224:2
11220:x
11217:b
11207:e
11203:)
11198:2
11194:C
11190:+
11187:x
11182:1
11178:C
11174:(
11171:=
11166:c
11162:y
11148:c
11144:b
11124:)
11118:c
11115:4
11107:2
11103:b
11094:b
11090:(
11083:2
11080:x
11071:e
11065:2
11061:C
11057:+
11051:)
11045:c
11042:4
11034:2
11030:b
11024:+
11021:b
11017:(
11010:2
11007:x
10998:e
10992:1
10988:C
10984:=
10979:c
10975:y
10961:c
10957:b
10939:p
10935:y
10931:+
10926:c
10922:y
10918:=
10915:y
10902:x
10900:(
10898:r
10889:p
10887:y
10869:x
10864:j
10855:e
10840:,
10838:e
10833:c
10831:y
10825:c
10823:y
10806:)
10803:x
10800:(
10797:r
10794:=
10791:y
10788:c
10785:+
10779:x
10776:d
10771:y
10768:d
10762:b
10759:+
10751:2
10747:x
10743:d
10738:y
10733:2
10729:d
10699:]
10693:2
10689:C
10685:+
10682:x
10677:1
10673:C
10669:+
10663:d
10659:)
10652:d
10648:)
10642:(
10639:Q
10631:d
10627:)
10621:(
10618:P
10604:e
10589:(
10583:x
10574:[
10565:d
10561:)
10555:(
10552:P
10547:x
10535:e
10531:=
10528:y
10502:d
10498:)
10492:(
10489:P
10484:x
10475:e
10451:)
10448:x
10445:(
10442:q
10439:=
10436:y
10432:)
10428:)
10425:x
10422:(
10415:p
10411:+
10406:2
10402:)
10398:x
10395:(
10392:p
10388:(
10384:+
10378:x
10375:d
10370:y
10367:d
10361:)
10358:x
10355:(
10352:p
10349:2
10346:+
10338:2
10334:x
10330:d
10325:y
10320:2
10316:d
10285:]
10281:C
10278:+
10272:d
10268:)
10262:(
10259:Q
10251:d
10247:)
10241:(
10238:P
10224:e
10218:x
10209:[
10200:d
10196:)
10190:(
10187:P
10182:x
10170:e
10166:=
10163:y
10140:.
10132:d
10128:)
10122:(
10119:P
10114:x
10105:e
10081:)
10078:x
10075:(
10072:Q
10069:=
10066:y
10063:)
10060:x
10057:(
10054:P
10051:+
10045:x
10042:d
10037:y
10034:d
10002:n
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9972:+
9964:1
9960:C
9956:+
9950:d
9946:)
9940:(
9937:F
9924:2
9916:d
9908:y
9897:=
9894:x
9882:,
9868:2
9863:)
9857:x
9854:d
9849:y
9846:d
9840:(
9832:x
9829:d
9825:d
9820:=
9812:2
9808:x
9804:d
9799:y
9794:2
9790:d
9780:x
9777:d
9772:y
9769:d
9763:2
9749:/
9745:2
9728:)
9725:y
9722:(
9719:F
9716:=
9708:2
9704:x
9700:d
9695:y
9690:2
9686:d
9633:d
9629:)
9623:,
9620:x
9617:(
9614:N
9611:)
9605:,
9602:x
9599:(
9591:y
9580:x
9565:)
9562:y
9559:,
9556:x
9553:(
9550:M
9547:=
9544:)
9541:x
9538:(
9535:X
9512:d
9508:)
9505:y
9502:,
9496:(
9493:M
9490:)
9487:y
9484:,
9478:(
9470:x
9459:y
9444:)
9441:y
9438:,
9435:x
9432:(
9429:N
9426:=
9423:)
9420:y
9417:(
9414:Y
9388:C
9385:=
9379:d
9375:)
9369:(
9366:X
9361:x
9353:+
9347:d
9343:)
9337:,
9334:x
9331:(
9328:N
9325:)
9319:,
9316:x
9313:(
9305:y
9293:=
9283:d
9279:)
9273:(
9270:Y
9265:y
9257:+
9251:d
9247:)
9244:y
9241:,
9235:(
9232:M
9229:)
9226:y
9223:,
9217:(
9209:x
9197:=
9194:)
9191:y
9188:,
9185:x
9182:(
9179:F
9164:)
9162:y
9158:x
9156:(
9154:μ
9133:x
9125:)
9122:N
9116:(
9107:=
9101:y
9093:)
9090:M
9084:(
9067:)
9065:y
9061:x
9059:(
9057:μ
9034:x
9026:N
9011:y
9003:M
8971:0
8968:=
8961:x
8958:d
8954:)
8951:y
8948:,
8945:x
8942:(
8939:N
8936:+
8933:y
8930:d
8926:)
8923:y
8920:,
8917:x
8914:(
8911:M
8904:0
8901:=
8894:)
8891:y
8888:,
8885:x
8882:(
8879:N
8876:+
8870:x
8867:d
8862:y
8859:d
8853:)
8850:y
8847:,
8844:x
8841:(
8838:M
8802:d
8798:)
8792:,
8789:x
8786:(
8783:N
8778:y
8767:x
8752:)
8749:y
8746:,
8743:x
8740:(
8737:M
8734:=
8731:)
8728:x
8725:(
8722:X
8699:d
8695:)
8692:y
8689:,
8683:(
8680:M
8675:x
8664:y
8649:)
8646:y
8643:,
8640:x
8637:(
8634:N
8631:=
8628:)
8625:y
8622:(
8619:Y
8594:C
8591:=
8585:d
8581:)
8575:(
8572:X
8567:x
8559:+
8553:d
8549:)
8543:,
8540:x
8537:(
8534:N
8529:y
8521:=
8508:d
8504:)
8498:(
8495:Y
8490:y
8482:+
8476:d
8472:)
8469:y
8466:,
8460:(
8457:M
8452:x
8444:=
8437:)
8434:y
8431:,
8428:x
8425:(
8422:F
8390:x
8382:N
8373:=
8367:y
8359:M
8327:0
8324:=
8317:x
8314:d
8310:)
8307:y
8304:,
8301:x
8298:(
8295:N
8292:+
8289:y
8286:d
8282:)
8279:y
8276:,
8273:x
8270:(
8267:M
8260:0
8257:=
8250:)
8247:y
8244:,
8241:x
8238:(
8235:N
8232:+
8226:x
8223:d
8218:y
8215:d
8209:)
8206:y
8203:,
8200:x
8197:(
8194:M
8170:C
8162:M
8158:N
8139:]
8136:)
8130:(
8127:M
8121:)
8115:(
8112:N
8109:[
8098:d
8094:)
8088:(
8085:N
8077:y
8074:x
8066:=
8063:)
8060:x
8057:C
8054:(
8014:0
8011:=
8004:y
8001:d
7997:)
7994:y
7991:x
7988:(
7985:N
7982:x
7979:+
7976:x
7973:d
7969:)
7966:y
7963:x
7960:(
7957:M
7954:y
7947:0
7944:=
7934:x
7931:d
7926:y
7923:d
7916:)
7913:y
7910:x
7907:(
7904:N
7901:x
7898:+
7895:)
7892:y
7889:x
7886:(
7883:M
7880:y
7842:)
7836:(
7833:F
7825:d
7817:x
7813:/
7809:y
7801:=
7798:)
7795:x
7792:C
7789:(
7771:x
7767:u
7745:)
7740:x
7737:y
7732:(
7728:F
7725:=
7719:x
7716:d
7711:y
7708:d
7660:C
7657:=
7651:d
7647:)
7641:(
7638:Q
7633:x
7625:+
7619:d
7615:)
7609:(
7606:P
7601:y
7567:0
7564:=
7557:x
7554:d
7550:)
7547:x
7544:(
7541:Q
7538:+
7535:y
7532:d
7528:)
7525:y
7522:(
7519:P
7512:0
7509:=
7502:)
7499:x
7496:(
7493:Q
7490:+
7484:x
7481:d
7476:y
7473:d
7467:)
7464:y
7461:(
7458:P
7444:y
7440:x
7422:C
7419:+
7413:)
7407:(
7404:F
7396:d
7388:y
7380:=
7377:x
7365:F
7341:x
7338:d
7334:)
7331:y
7328:(
7325:F
7322:=
7315:y
7312:d
7305:)
7302:y
7299:(
7296:F
7293:=
7283:x
7280:d
7275:y
7272:d
7255:y
7237:C
7234:+
7228:d
7224:)
7218:(
7215:F
7210:x
7202:=
7199:y
7170:x
7167:d
7163:)
7160:x
7157:(
7154:F
7151:=
7144:y
7141:d
7134:)
7131:x
7128:(
7125:F
7122:=
7112:x
7109:d
7104:y
7101:d
7084:x
7066:C
7063:=
7057:d
7050:)
7044:(
7039:1
7035:Q
7029:)
7023:(
7018:2
7014:Q
7005:y
6997:+
6991:d
6984:)
6978:(
6973:2
6969:P
6963:)
6957:(
6952:1
6948:P
6939:x
6922:1
6919:Q
6916:2
6913:P
6892:0
6889:=
6882:y
6879:d
6875:)
6872:y
6869:(
6864:2
6860:Q
6856:)
6853:x
6850:(
6845:2
6841:P
6837:+
6834:x
6831:d
6827:)
6824:y
6821:(
6816:1
6812:Q
6808:)
6805:x
6802:(
6797:1
6793:P
6785:0
6782:=
6772:x
6769:d
6764:y
6761:d
6754:)
6751:y
6748:(
6743:2
6739:Q
6735:)
6732:x
6729:(
6724:2
6720:P
6716:+
6713:)
6710:y
6707:(
6702:1
6698:Q
6694:)
6691:x
6688:(
6683:1
6679:P
6663:y
6659:x
6632:x
6628:λ
6619:λ
6615:)
6613:λ
6611:(
6609:F
6599:λ
6597:(
6595:F
6576:2
6573:C
6569:1
6566:C
6560:c
6554:b
6548:y
6542:x
6533:)
6531:y
6529:,
6527:x
6525:(
6523:N
6518:)
6516:y
6514:,
6512:x
6510:(
6508:M
6503:)
6501:y
6499:(
6497:Q
6492:)
6490:y
6488:(
6486:P
6481:)
6479:x
6477:(
6475:Q
6470:)
6468:x
6466:(
6464:P
6438:.
6435:)
6432:)
6427:n
6423:y
6419:,
6413:,
6408:1
6404:y
6400:,
6397:x
6394:(
6391:F
6388:,
6383:n
6379:y
6375:,
6369:,
6364:2
6360:y
6356:(
6353:=
6350:)
6345:n
6341:y
6337:,
6331:,
6326:1
6322:y
6318:,
6315:x
6312:(
6308:F
6303:,
6300:)
6295:n
6291:y
6287:,
6281:,
6276:1
6272:y
6268:(
6265:=
6261:y
6234:)
6230:y
6226:,
6223:x
6220:(
6216:F
6212:=
6204:y
6172:.
6169:)
6164:n
6160:y
6156:,
6150:,
6145:1
6141:y
6137:,
6134:x
6131:(
6128:F
6123:=
6113:n
6109:y
6099:n
6095:y
6089:=
6079:1
6073:n
6069:y
6050:3
6046:y
6040:=
6030:2
6026:y
6016:2
6012:y
6006:=
5996:1
5992:y
5974:n
5970:n
5966:i
5948:.
5943:)
5940:1
5934:i
5931:(
5927:y
5923:=
5918:i
5914:y
5900:n
5881:)
5878:n
5875:(
5871:y
5867:=
5863:)
5857:)
5854:1
5848:n
5845:(
5841:y
5834:,
5825:,
5818:y
5814:,
5807:y
5803:,
5800:y
5797:,
5794:x
5790:(
5786:F
5773:n
5736:,
5724:)
5721:x
5718:(
5715:y
5701:x
5694:x
5665:R
5640:,
5637:)
5632:0
5628:y
5623:/
5619:1
5616:+
5611:0
5607:x
5603:(
5596:R
5585:y
5561:0
5553:0
5549:y
5542:)
5535:+
5532:,
5525:0
5521:y
5517:1
5512:+
5507:0
5503:x
5498:(
5490:0
5482:0
5478:y
5471:)
5463:0
5459:y
5455:1
5450:+
5445:0
5441:x
5437:,
5427:(
5419:0
5416:=
5411:0
5407:y
5400:R
5393:{
5362:1
5359:+
5354:0
5350:y
5346:)
5343:x
5335:0
5331:x
5327:(
5321:0
5317:y
5311:=
5308:)
5305:x
5302:(
5299:y
5275:R
5261:C
5257:y
5251:(
5249:F
5230:2
5226:y
5222:=
5215:y
5195:0
5192:y
5188:0
5185:x
5168:R
5115:F
5077:)
5074:x
5071:(
5068:y
5057:x
5050:x
5014:)
5011:x
5008:(
5005:y
4991:x
4984:x
4940:x
4910:I
4876:I
4867:0
4863:x
4859:,
4856:}
4847:{
4840:R
4827:x
4823:,
4820:)
4815:+
4811:x
4807:,
4798:x
4794:(
4791:=
4782:I
4768:0
4765:y
4761:0
4758:x
4743:y
4739:x
4737:(
4735:F
4731:F
4722:R
4718:h
4703:]
4700:a
4697:+
4692:0
4688:x
4684:,
4681:a
4673:0
4669:x
4665:[
4659:]
4656:h
4653:+
4648:0
4644:x
4640:,
4637:h
4629:0
4625:x
4621:[
4618:=
4615:I
4597:×
4592:R
4588:b
4584:a
4575:b
4571:a
4553:]
4550:b
4547:+
4542:0
4538:y
4534:,
4531:b
4523:0
4519:y
4515:[
4509:]
4506:a
4503:+
4498:0
4494:x
4490:,
4487:a
4479:0
4475:x
4471:[
4468:=
4465:R
4455:y
4451:F
4447:F
4433:)
4428:0
4424:x
4420:(
4417:y
4414:=
4409:0
4405:y
4400:,
4396:)
4393:y
4390:,
4387:x
4384:(
4381:F
4378:=
4371:y
4179:f
4132:n
4070:2
4065:)
4060:|
4054:2
4051:x
4043:1
4039:|
4035:+
4030:2
4027:x
4019:1
4015:(
4008:4
4005:1
4000:=
3997:)
3994:x
3991:(
3988:y
3962:1
3959:=
3956:)
3953:0
3950:(
3947:y
3942:,
3936:|
3932:y
3928:|
3922:)
3919:y
3916:(
3905:=
3898:y
3829:n
3825:n
3810:R
3788:.
3785:I
3779:x
3775:)
3772:x
3769:(
3766:v
3763:=
3760:)
3757:x
3754:(
3751:u
3737:J
3733:I
3728:v
3720:u
3715:R
3711:R
3707:I
3703:v
3697:R
3693:R
3689:J
3685:u
3667:.
3664:I
3658:x
3654:0
3651:=
3648:)
3643:)
3640:n
3637:(
3633:u
3626:,
3617:,
3610:u
3606:,
3603:u
3600:,
3597:x
3594:(
3591:F
3578:I
3574:n
3570:u
3566:F
3554:I
3549:R
3545:R
3541:I
3537:u
3519:0
3516:=
3512:)
3506:)
3503:n
3500:(
3496:y
3492:,
3486:,
3479:y
3475:,
3472:y
3469:,
3466:x
3462:(
3458:F
3399:v
3390:)
3386:v
3382:,
3378:u
3374:,
3371:x
3368:(
3364:F
3333:0
3329:=
3325:)
3316:y
3311:,
3307:y
3303:,
3300:x
3296:(
3291:F
3265:)
3259:0
3245:0
3238:0
3232:(
3227:=
3222:)
3216:)
3211:)
3208:n
3205:(
3200:y
3195:,
3189:,
3181:y
3176:,
3168:y
3163:,
3159:y
3155:,
3152:x
3149:(
3144:m
3140:f
3125:)
3120:)
3117:n
3114:(
3109:y
3104:,
3098:,
3090:y
3085:,
3077:y
3072:,
3068:y
3064:,
3061:x
3058:(
3053:2
3049:f
3041:)
3036:)
3033:n
3030:(
3025:y
3020:,
3014:,
3006:y
3001:,
2993:y
2988:,
2984:y
2980:,
2977:x
2974:(
2969:1
2965:f
2958:(
2939:0
2921:0
2917:=
2913:)
2907:)
2904:n
2901:(
2896:y
2891:,
2885:,
2877:y
2872:,
2864:y
2859:,
2855:y
2851:,
2848:x
2844:(
2839:F
2806:)
2799:)
2793:)
2790:1
2784:n
2781:(
2776:y
2771:,
2765:,
2757:y
2752:,
2744:y
2739:,
2735:y
2731:,
2728:x
2724:(
2718:m
2714:f
2698:)
2692:)
2689:1
2683:n
2680:(
2675:y
2670:,
2664:,
2656:y
2651:,
2643:y
2638:,
2634:y
2630:,
2627:x
2623:(
2617:2
2613:f
2604:)
2598:)
2595:1
2589:n
2586:(
2581:y
2576:,
2570:,
2562:y
2557:,
2549:y
2544:,
2540:y
2536:,
2533:x
2529:(
2523:1
2519:f
2512:(
2507:=
2502:)
2494:)
2491:n
2488:(
2483:m
2479:y
2462:)
2459:n
2456:(
2451:2
2447:y
2437:)
2434:n
2431:(
2426:1
2422:y
2415:(
2396:m
2389:n
2364:)
2358:)
2355:1
2349:n
2346:(
2341:y
2336:,
2330:,
2322:y
2317:,
2309:y
2304:,
2300:y
2296:,
2293:x
2289:(
2284:F
2280:=
2275:)
2272:n
2269:(
2264:y
2249:y
2241:F
2237:x
2235:(
2233:y
2229:y
2197:x
2195:(
2193:r
2176:.
2172:y
2161:x
2159:(
2157:r
2133:x
2131:(
2129:r
2125:x
2121:)
2119:x
2115:r
2110:)
2108:x
2106:(
2100:i
2095:a
2078:)
2075:x
2072:(
2069:r
2066:+
2061:)
2058:i
2055:(
2051:y
2047:)
2044:x
2041:(
2036:i
2032:a
2026:1
2020:n
2015:0
2012:=
2009:i
2001:=
1996:)
1993:n
1990:(
1986:y
1972:y
1953:F
1931:.
1928:x
1895:0
1892:=
1888:)
1882:)
1879:n
1876:(
1872:y
1865:,
1856:,
1849:y
1845:,
1838:y
1834:,
1831:y
1828:,
1825:x
1821:(
1817:F
1804:n
1771:)
1768:n
1765:(
1761:y
1757:=
1753:)
1747:)
1744:1
1738:n
1735:(
1731:y
1727:,
1721:,
1714:y
1710:,
1707:y
1704:,
1701:x
1697:(
1693:F
1680:y
1676:y
1672:x
1668:F
1640:)
1634:.
1631:.
1628:.
1624:y
1619:,
1610:y
1604:,
1595:y
1589:(
1572:)
1570:y
1566:y
1562:y
1560:(
1549:)
1539:/
1535:y
1532:d
1516:/
1512:y
1509:d
1493:/
1483:(
1472:x
1465:)
1463:x
1461:(
1459:f
1455:y
1446:y
1434:t
1432:(
1430:x
1428:(
1426:F
1422:t
1420:(
1418:x
1414:t
1410:t
1408:(
1406:x
1402:F
1398:m
1376:)
1373:)
1370:t
1367:(
1364:x
1361:(
1358:F
1355:=
1347:2
1343:t
1338:d
1332:)
1329:t
1326:(
1323:x
1318:2
1313:d
1305:m
1292:F
1288:t
1284:x
1133:x
1129:y
1111:)
1108:n
1105:(
1101:y
1097:,
1091:,
1084:y
1055:)
1052:x
1049:(
1046:b
1022:)
1019:x
1016:(
1011:n
1007:a
982:)
979:x
976:(
971:0
967:a
941:,
938:0
935:=
932:)
929:x
926:(
923:b
920:+
915:)
912:n
909:(
905:y
901:)
898:x
895:(
890:n
886:a
882:+
876:+
869:y
865:)
862:x
859:(
854:2
850:a
846:+
839:y
835:)
832:x
829:(
824:1
820:a
816:+
813:y
810:)
807:x
804:(
799:0
795:a
729:(
710:e
703:t
696:v
509:)
505:(
20:)
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